Sigmatropic Rearrangements: Order and Nomenclature

Sigmatropic rearrangements are not museum pieces confined to physical organic chemistry courses — they are workhorse steps in modern total synthesis and industrial process chemistry precisely because they form carbon–carbon and carbon–heteroatom bonds with predictable stereochemistry and, crucially, without needing an external reagent or metal catalyst to do so. The Ireland–Claisen rearrangement, a $[3,3]$-shift of a silyl ketene acetal derived from an allylic ester, is a standard method for constructing quaternary stereocenters with defined relative configuration in the synthesis of terpenoids and polyketide natural products, because the chair-like transition-state geometry translates ester enolate geometry directly into product stereochemistry with high fidelity. The anionic oxy-Cope rearrangement, a $[3,3]$-shift dramatically accelerated (by factors of $10^{10}$–$10^{17}$ relative to the neutral oxy-Cope) when the migrating alkoxide is deprotonated, is a key strategic disconnection in the synthesis of complex polycyclic terpenoids, because it converts an easily made 1,5-hexadien-3-ol into a ring-expanded or ring-rearranged ketone in one step under mild conditions, a transformation exploited extensively in steroid and terpenoid total synthesis programs. In biochemistry, chorismate mutase catalyzes a $[3,3]$-sigmatropic rearrangement — chorismate to prephenate — that is the committed step toward phenylalanine and tyrosine biosynthesis in plants, fungi, and bacteria (but conspicuously absent in mammals, which obtain these amino acids only from diet). Because this rearrangement is entirely absent from human metabolism yet essential to many pathogens, chorismate mutase is an actively pursued target for antimicrobial and antifungal drug design — inhibitors designed to mimic the pericyclic chair-like transition state of the uncatalyzed rearrangement have been used as mechanistic probes and lead compounds. $[1,5]$-hydrogen shifts govern the thermal behavior of cyclopentadiene derivatives (rapid tautomer scrambling observed by variable-temperature NMR) and are implicated in the biosynthesis and photochemical interconversion of vitamin D isomers, where a series of electrocyclic and sigmatropic steps convert 7-dehydrocholesterol, via previtamin D$_3$, into the final vitamin D$_3$ structure through a thermal $[1,7]$-H shift — an unusually high-order sigmatropic shift made possible by the extended triene conjugation in the secosteroid intermediate. The $[2,3]$-Wittig rearrangement and its nitrogen and sulfur ylide analogues (Stevens and Sommelet–Hauser rearrangements) remain standard disconnections for constructing homoallylic alcohols and functionalized amines with predictable stereochemical transfer from an existing stereocenter to the newly formed bond, a strategy used throughout process-scale pharmaceutical synthesis when chiral-pool starting materials are available.

:::quiz[What determines the two numbers i and j in an [i,j]-sigmatropic shift?]

• The total number of π electrons in the whole starting material
• The number of atoms counted outward from the breaking σ bond along each fragment ✓
• The number of substituents on the migrating group
• The overall symmetry point group of the transition state

:::quiz[A [2,3]-Wittig rearrangement transition state ring contains how many atoms?]

• 3
• 4
• 5 ✓
• 6

:::quiz[A [1,5]-hydrogen shift and a [3,3]-shift both proceed through transition states with how many total atoms, despite differing atom distribution between fragments?]

• 4 atoms in both
• 5 atoms in both
• 6 atoms in both ✓
• Different totals; they cannot be compared

:::faq[Is a [1,1]-shift a real thing?] Formally you could write a $[1,1]$ label for a process where a group returns to essentially the same position, but this describes no bond change at all (position 1 to position 1 is the original bond), so it is never used as a meaningful classification — the smallest chemically meaningful sigmatropic shifts are $[1,3]$ (four-atom transition state) for simple allylic migrations. There is no lower nontrivial member of the family, because you need at least one atom of "reach" beyond the original bond on at least one fragment for any bond reorganization to have actually occurred.

:::faq[Do both fragments in a [3,3]-shift have to be carbon allyl systems, or can one be a heteroatom-containing system?] Either or both fragments can contain heteroatoms — the counting procedure is completely agnostic to atomic identity. The Claisen rearrangement is the classic example where one fragment is an all-carbon allyl group and the other includes an oxygen at position 1 or 2 of its numbering (oxygen is typically at position 1, directly part of the original $\sigma$ bond, with the vinyl carbons at positions 2 and 3). Aza-Cope rearrangements place a nitrogen in an analogous position, and thio-Claisen rearrangements use sulfur. In every case you renumber starting from the actual breaking bond and count outward exactly as with an all-carbon system.

:::faq[How does the [i,j] label relate to whether a shift is thermally allowed?] The order determines the total electron count in the pericyclic array (via $N_\pi = i+j-2$, plus the two electrons of the migrating bond itself, giving a total of $i+j$ electrons distributed over $i+j$ atoms in most simple cases). Woodward–Hoffmann analysis, covered in the Woodward-Hoffmann Rules and Orbital Symmetry framework, then asks whether that electron count is $4n$ or $4n+2$ and whether the process is suprafacial or antarafacial on each fragment; $4n+2$ electron systems favor an all-suprafacial thermal pathway (like $[3,3]$ and $[1,5]$ shifts, both 6-electron systems), while $4n$ systems (like a naive all-suprafacial $[1,3]$-H shift, a 4-electron system) are thermally forbidden in the simple suprafacial-suprafacial sense and require either photochemical activation or a geometrically awkward antarafacial component.

:::faq[Why does hydrogen almost never do a suprafacial [1,3]-shift thermally, while carbon sometimes can?] A thermal $[1,3]$-shift (4 electrons, $4n$ count) is Woodward–Hoffmann allowed only if one component is suprafacial and the other is antarafacial. For a migrating hydrogen, "antarafacial" would require the hydrogen's 1s orbital to bond to the opposite face of the allyl system from which it departed — geometrically essentially impossible for a single spherical s-orbital with no second lobe to invert into, since hydrogen has no $p$-orbital character to flip phase. A migrating carbon, however, can be antarafacial at itself by inverting its configuration (like an SN2-type backside process) while retaining a suprafacial allyl component, which is why authenticated thermal $[1,3]$-carbon shifts proceeding with inversion at the migrating carbon are well documented, whereas thermal suprafacial $[1,3]$-H shifts are essentially never observed without photochemical or catalytic assistance.

:::faq[Is the [i,j] order affected by whether the reaction is thermal or photochemical?] No — the order label is a purely structural/connectivity statement about which atoms bond to which before and after rearrangement; it says nothing about the mechanism's electron count parity or the reaction conditions used to achieve it. The same $[i,j]$ label can in principle describe both a thermally allowed pathway (all-suprafacial for a 6-electron system) and, under different substitution or forcing conditions, a photochemically driven variant of a normally 4-electron-forbidden thermal case. Classify the order first, purely from structure; only afterward do you layer on the Woodward–Hoffmann electron-counting analysis to decide how (suprafacial/antarafacial, thermal/photochemical) the shift can actually occur.

Ask the tutor: How do the suprafacial and antarafacial descriptors combine with an [i,j] label to fully specify a sigmatropic rearrangement's stereochemical outcome?

Picture the migrating $\sigma$ bond as a tether connecting two ends of a floppy, partially unsaturated chain. In a sigmatropic rearrangement, one end of the tether detaches from its atom and a new tether forms further along the chain, while the $\pi$ bonds shuffle by exactly one position to accommodate the new connectivity — like a caterpillar's body rippling forward as it walks, except here it is bonding electrons that ripple rather than the whole molecule translating in space. Start with the simplest case: an allylic system with a substituent $X$ attached at C1 of a three-carbon allyl unit, C1–C2=C3. The $\sigma$ bond is C1–X. If $X$ migrates from C1 directly to C3 (the far end of the allyl system), the $\pi$ bond must shift from C2=C3 to C1=C2 to conserve valence. This is the archetype $[1,3]$-shift: count "1" from the X atom outward (X is atom 1 on its own fragment, since it does not extend further — X has nowhere else to go, so its count stops at 1), and count from C1 (also labeled atom 1, the carbon originally bonded to X) through C2 to C3 (atom 3) on the other fragment. Hence "$[1,3]$": one atom on the X-side, three atoms on the carbon-side, and the new bond forms between position 1 of the first fragment and position 3 of the second. Now extend the unsaturation: if instead of a single allyl unit we have a pentadienyl framework, C1–C2=C3–C4=C5, and $X$ migrates all the way from C1 to C5, we get a $[1,5]$-shift — still one atom on the X-fragment, but five atoms on the carbon fragment, with both $\pi$ bonds shuffling forward by one position (C2=C3 and C4=C5 become C1=C2 and C3=C4). The transition-state ring in this case has $1+5=6$ atoms, exactly the same ring size as a $[3,3]$-shift, but the atom count on each side is different, which changes which orbitals must overlap and hence the suprafacial/antarafacial requirements analyzed via the Woodward-Hoffmann Rules and Orbital Symmetry. The $[3,3]$-shift is different in kind: here both fragments are allylic. Take 1,5-hexadiene: C1=C2–C3–C4–C5=C6. The $\sigma$ bond that migrates is the central C3–C4 bond. Splitting the molecule at that bond gives two allyl fragments, C3–C2=C1 and C4–C5=C6. Renumber each fragment starting at the breaking bond: on the left fragment C3 is atom 1, C2 is atom 2, C1 is atom 3; on the right fragment C4 is atom 1, C5 is atom 2, C6 is atom 3. The new $\sigma$ bond forms between atom 3 of the left fragment (former C1) and atom 3 of the right fragment (former C6) — a $[3,3]$-shift, with both alkenes sliding inward by one position (C1=C2 becomes C2=C3, and C5=C6 becomes C4=C5).

:::svg[Counting the order of a [3,3]-shift in 1,5-hexadiene: renumber each fragment from the breaking σ bond outward]

::: The picture generalizes cleanly: whatever the substituent pattern, the recipe is always (1) find the $\sigma$ bond that breaks, (2) split the molecule mentally at that bond into two fragments, (3) number each fragment independently starting at "1" for the atom that was part of the breaking bond, and (4) walk outward along each fragment counting atoms until you reach the atom that forms the new bond. Whatever two numbers you land on are $i$ and $j$.

:::svg[[1,5]-H shift in (Z)-1,3-pentadiene: one atom migrates across a four-carbon π framework]

::: Notice the asymmetry built into the $[1,5]$ case versus the $[3,3]$ case: both have six-atom transition states, but a $[1,5]$-shift places a single atom (often hydrogen, which has no other substituents to worry about, hence no stereochemical descriptor on that fragment) at one terminus, while a $[3,3]$-shift splits the six atoms 3-and-3, so that stereocenters or double-bond geometry on both halves matter for the product outcome. This distinction becomes essential once you analyze suprafacial versus antarafacial pathways, because the number of atoms in each fragment determines how many nodes are present in the relevant frontier orbital of that fragment, and hence which face of that fragment participates in bonding.

The formal definition, stated as a single relationship, is this: given a $\sigma$ bond joining atom $A$ to atom $A'$, flanked by a $\pi$ system that permits the bond to migrate, renumber each side of the molecule starting from the breaking bond as position 1, and let the new $\sigma$ bond form between position $i$ on one fragment and position $j$ on the other. Then the process is a $[i,j]$-sigmatropic rearrangement.

where: $A_1$ and $A_1'$ are the two atoms of the original $\sigma$ bond (subscript 1 by convention on each side); $A_i$ is the atom $i-1$ positions away from $A_1$ along one $\pi$-conjugated fragment; $A_j'$ is the atom $j-1$ positions away from $A_1'$ along the other fragment; the new bond is $A_i$–$A_j'$. The transition-state ring size follows immediately:

where: $n_{\text{TS ring}}$ is the number of atoms in the cyclic array of overlapping orbitals during the concerted shift (all atoms of both fragments up to and including the migration termini, i.e. positions $1,2,\dots,i$ on one side and $1,2,\dots,j$ on the other, but position 1 is shared conceptually as the "hinge" of the original bond so the total atom count in the ring is exactly $i+j$, not $i+j-1$, because the two "position 1" atoms are different atoms joined by the bond that is breaking). A useful auxiliary count is the number of $\pi$ electrons formally engaged in the pericyclic array, since Woodward–Hoffmann analysis is normally phrased in terms of total electron count:

where: $N_\pi$ is the number of $\pi$/$\sigma$-bonding electron pairs in the cyclic transition state beyond the migrating $\sigma$ bond's own electron pair (this counts the $\pi$ bonds that participate — a $[3,3]$-shift, $i=j=3$, gives $N_\pi = 4$ electrons' worth of $\pi$ participation across the two allyl fragments, consistent with the familiar "6-electron, 6-membered, aromatic-TS" description of the Cope rearrangement when the migrating $\sigma$ electron pair is included to make the total 6 electrons over 6 atoms). Reading the second relationship: for a $[1,3]$-shift, $N_\pi = 1+3-2 = 2$, meaning one alkene ($\pi$ bond, 2 electrons) is engaged besides the migrating bond — matches a simple allylic system. For a $[3,3]$-shift, $N_\pi = 3+3-2 = 4$, meaning two alkenes are engaged, matching the two allyl fragments of hexadiene. For a $[1,5]$-shift, $N_\pi = 1+5-2 = 4$, again two alkenes engaged — a pentadienyl system has two $\pi$ bonds — even though the atom split (1 vs 5) is very different from the $[3,3]$ case (3 vs 3). This is exactly why $[1,5]$-H shifts and $[3,3]$-shifts are both thermally allowed suprafacial-suprafacial processes: they have the same total electron count (6, once the migrating bond pair is added to $N_\pi$), even though the atom distribution between fragments differs.

:::derivation[Deriving the [i,j] label mechanistically for an allylic system, atom by atom]

Goal. Starting from an explicit curved-arrow mechanism for the migration of a group $X$ across an allyl fragment, derive the $[1,3]$ label from first principles rather than by pattern-matching a memorized picture.

Step 1 — Identify the σ bond and draw it as bond 1–1′. Consider 3-substituted-1-propene bearing a leaving-capable $\sigma$ substituent $X$ at the sp$^3$ carbon: $X$–CH$_2$–CH=CH$_2$. Label the carbon bonded to $X$ as $C_a$ and call this pair the origin of numbering: $X$ is atom "1" on its own fragment (it has no further atoms to traverse, since $X$ is the migrating group, not part of the $\pi$ conveyor), and $C_a$ is atom "1" on the carbon fragment. Write the $\sigma$ bond explicitly as $X_1$–$(C_a)_1$.

Step 2 — Number outward along the π-conjugated fragment. From $C_a$ (position 1), move along the chain: the next atom is the alkene carbon $C_b$ (position 2), and the terminal alkene carbon $C_c$ is position 3. The $\pi$ bond in the starting material sits between position 2 and position 3 ($C_b=C_c$).

Step 3 — Draw the curved arrows for the concerted shift. Three curved arrows describe the pericyclic flow: arrow 1 breaks the $X$–$C_a$ $\sigma$ bond, with the electron pair moving to become the new $\pi$ bond between $C_a$ and $C_b$ (positions 1 and 2); arrow 2 shifts the existing $\pi$ bond ($C_b=C_c$, positions 2–3) so that one electron pair moves to form the new $\sigma$ bond between $X$ and $C_c$ (position 3). The array closes into a five-membered cyclic arrow pattern (counting $X$, $C_a$, $C_b$, $C_c$, and back to $X$) — but note the atom count, not the arrow count, is what defines the order.

Step 4 — Read off the order. $X$ never advances past position 1 on its own fragment (it terminates the chain). The carbon fragment advances from position 1 ($C_a$, where the old bond was) to position 3 ($C_c$, where the new bond forms). Hence the shift is labeled using the pair (1, 3): a $[1,3]$-shift. The transition-state ring contains $1+3=4$... but wait — check this against the explicit atoms drawn: $X$, $C_a$, $C_b$, $C_c$ is four atoms total, and $1+3=4$. The formula $n_{\text{TS ring}}=i+j$ is confirmed directly by counting the atoms drawn in the cyclic arrow array.

Step 5 — Sanity-check with the electron count. $N_\pi = i+j-2 = 1+3-2 = 2$, i.e., exactly one alkene ($C_b=C_c$) participates beyond the migrating bond's own electron pair — consistent with the single $\pi$ bond actually present in the allyl fragment before rearrangement. This cross-check (electron count from the formula versus electron count read directly off the structure) is the standard way to confirm a proposed order is self-consistent.

:::derivation[Deriving the order for an unsymmetrical, asymmetric-substituted [2,3]-Wittig-type rearrangement]

Goal. Apply the same counting procedure to a case where both fragments extend more than one atom but by different amounts, using an unsymmetrically substituted ether-ylide system — the classic setup for a [2,3]-Wittig rearrangement — to show the method handles heteroatoms and anionic character without modification.

Step 1 — Draw the substrate and locate the σ bond that migrates. Take an allylic ether bearing a carbanion generated alpha to oxygen on the other alkyl arm: $R$–CH(Li)–O–CH$_2$–CH=CH$_2$, where deprotonation alpha to oxygen (by a strong base such as n-BuLi) has generated a stabilized carbanion at the carbon labeled $C_x$ ($R$–$\overset{-}{C}$H–O–…). The $\sigma$ bond that will migrate is the O–CH$_2$ bond connecting the ether oxygen to the allylic carbon; call the oxygen atom $O_1$ and the allylic carbon it is bonded to $C_1'$ on the other fragment.

Step 2 — Number the carbanion fragment. Starting the count at the atom bonded to oxygen: the oxygen itself is atom "1" on its own fragment side (position 1, the anionic fragment terminates the count here structurally because the carbanion lone pair, not a further chain atom, is what attacks) — more precisely, in the standard convention for $[2,3]$-Wittig nomenclature, the carbanion carbon $C_x$ is counted as position 1, and the oxygen is position 2 (the carbanion pushes through the oxygen lone pair's adjacent bond framework is not how this works — instead $C_x$(1)–$O$(2) is the two-atom fragment, because the carbanion lone pair is the nucleophile and the oxygen is the atom directly bonded to the departing allylic system).

Step 3 — Number the allyl fragment. On the other side, the allylic system: $C_1'$ (the carbon originally bonded to O, i.e. the allylic CH$_2$), $C_2'$ (the internal alkene carbon), $C_3'$ (the terminal alkene carbon). The new bond forms between the carbanion carbon and $C_3'$.

Step 4 — Assemble the curved-arrow flow. Arrow 1: the carbanion lone pair on $C_x$ attacks the terminal alkene carbon $C_3'$, forming the new C–C $\sigma$ bond. Arrow 2: the $\pi$ bond $C_2'=C_3'$ shifts to become the new $\pi$ bond $C_1'=C_2'$. Arrow 3: the O–$C_1'$ $\sigma$ bond breaks heterolytically, with the electron pair returning to oxygen as a lone pair (regenerating a neutral alkoxide oxygen, i.e. the product is an alkoxide — this rearrangement converts a carbanion + ether into an alkoxide + new C–C bond, which is the defining feature of the Wittig rearrangement family).

Step 5 — Read off the order and ring size. Fragment one (carbanion side): $C_x$ = 1, $O$ = 2 — two atoms. Fragment two (allyl side): $C_1'=1$, $C_2'=2$, $C_3'=3$ — three atoms. The shift is therefore $[2,3]$, and the pericyclic transition-state ring has $n_{\text{TS ring}} = 2+3 = 5$ atoms: $C_x$, $O$, $C_1'$, $C_2'$, $C_3'$, closing back to $C_x$. The electron count $N_\pi = 2+3-2=3$ pairs beyond the forming bond is consistent with one $\pi$ bond (the allyl alkene, 2 electrons) plus the carbanion lone pair (2 electrons) forming the new bond, totaling the expected six-electron, five-membered aromatic-like transition state once all contributing electron pairs are tallied consistently with the Woodward-Hoffmann Rules and Orbital Symmetry framework for $4n+2$ suprafacial-suprafacial thermal pathways.

Example 1 — Foundational: classify the Claisen rearrangement of allyl vinyl ether. Allyl vinyl ether is CH$_2$=CH–O–CH$_2$–CH=CH$_2$. Identify the migrating $\sigma$ bond: it is the O–CH$_2$(allyl) bond, the only $\sigma$ bond directly flanked on both sides by a $\pi$ system (the vinyl ether alkene on one side of oxygen, the allyl alkene on the other side of the CH$_2$). Split the molecule at that bond. Fragment A (vinyl ether side): oxygen is bonded to the vinyl group. Number starting at the oxygen atom itself as position 1, then the adjacent vinyl carbon as position 2, then the terminal vinyl carbon as position 3: $O(1)$–$CH(2)$=$CH_2(3)$. Fragment B (allyl side): the carbon originally bonded to oxygen is position 1, the internal alkene carbon is position 2, the terminal alkene carbon is position 3: $CH_2(1)$–$CH(2)$=$CH_2(3)$. The new bond forms between position 3 of fragment A (the terminal vinyl carbon, which becomes a new carbonyl carbon after tautomerization... actually becomes an enol carbon directly, which tautomerizes fast to the aldehyde) and position 3 of fragment B (the terminal allyl carbon). This is a $[3,3]$-shift, transition-state ring size $3+3=6$ atoms, and the direct product is 4-pentenal after the initially formed enol tautomerizes. Checking the electron count: $N_\pi = 3+3-2=4$, i.e. two $\pi$ bonds engaged (the vinyl ether alkene and the allyl alkene) plus the migrating $\sigma$ bond's electrons, totaling six electrons over six atoms — the canonical aromatic transition state of the Claisen rearrangement.

Example 2 — Applied: classify a base-mediated [2,3]-shift in a sulfonium ylide (Sommelet–Hauser-type counting) with asymmetric substitution. Consider an allylic sulfonium salt, $\mathrm{PhCH_2}$–$\overset{+}{S}(\mathrm{Me})$–$\mathrm{CH_2CH{=}CHCH_3}$ (note the unsymmetrical, non-terminal alkene — a but-2-enyl group rather than a simple allyl group, chosen deliberately so the two ends of the allyl fragment are inequivalent). Treatment with base deprotonates the benzylic C–H alpha to sulfur, generating an ylide: $\mathrm{Ph\overset{-}{C}H}$–$\overset{+}{S}(\mathrm{Me})$–$\mathrm{CH_2CH{=}CHCH_3}$. Identify the migrating $\sigma$ bond: the S–CH$_2$ bond connecting sulfur to the crotyl (but-2-enyl) group is flanked by the alkene on one side and by sulfur (bearing the adjacent ylide carbon) on the other — this is the bond that breaks. Number fragment A (ylide side): carbanion carbon is position 1 ($C_x$, bearing the phenyl and the lone pair that will attack), sulfur is position 2. Two atoms: $C_x(1)$–$S(2)$. Number fragment B (crotyl side): the carbon originally bonded to sulfur is position 1, the internal alkene carbon is position 2, and — because this is a but-2-enyl group, not a simple allyl — the far alkene carbon (bearing the extra methyl substituent) is position 3: $CH_2(1)$–$CH(2)$=$CH(3)$(CH$_3$). The new C–C bond forms between the ylide carbon (position 1 of fragment A) and the far, methyl-bearing alkene carbon (position 3 of fragment B). This is a $[2,3]$-shift exactly as before — the presence of the extra methyl substituent on the terminal alkene carbon changes the regiochemistry and stereochemistry of the product (it introduces a new stereocenter or alters alkene geometry in the product) but does not change the order label, because the order counts atoms along the chain, not substituents hanging off those atoms. This is a critical, exam-relevant distinction: substituents are irrelevant to $[i,j]$ classification; only the connectivity backbone matters. Ring size: $2+3=5$ atoms ($C_x$, $S$, $CH_2$, $CH$, $CH$), matching the Sommelet–Hauser $[2,3]$-sigmatropic shift used to convert benzylic ammonium/sulfonium ylides into ring-substituted (ortho) products via rearomatization, a classic route in heteroatom-directed [2,3] chemistry.

Example 3 — Exam/competition level: a substrate with two candidate σ bonds, only one of which is sigmatropically allowed. A compound has the connectivity $\mathrm{CH_2{=}CH{-}CH_2{-}O{-}CH_2{-}CH{=}CH_2}$ (an allyl allyl ether — allyl group on each side of oxygen, but written out fully to force careful bond-counting) alongside a second, non-conjugated $\sigma$ bond: a methyl group attached to the central oxygen-bearing region is not present here, so instead let's sharpen the challenge by using allyl propargyl ether: $\mathrm{CH_2{=}CH{-}CH_2{-}O{-}CH_2{-}C{\equiv}CH}$. There are, in principle, two $\sigma$ bonds flanked by $\pi$ systems on both sides that could be candidates for a sigmatropic shift: the O–CH$_2$(allyl) bond and the O–CH$_2$(propargyl) bond. Only one of these is "the" bond that migrates in the thermal $[3,3]$ rearrangement of this substrate (the propargyl Claisen rearrangement), and determining which one requires checking that both fragments generated by cleavage are themselves properly conjugated all the way to a terminus capable of accepting the new bond. Take the O–CH$_2$(allyl) bond as the candidate breaking bond. Fragment A (across the allyl side): $O(1)$–$CH_2(2)$... wait — check this numbering against the earlier convention: the oxygen itself, being directly bonded to the other $\pi$ system (the propargyl side) via a second $\sigma$ bond, is not itself part of a $\pi$-conjugated chain on the allyl fragment; rather, oxygen bonded to allyl needs the allyl-side $\pi$ system counted starting from oxygen. Redo carefully: Fragment A is the O–allyl piece: $O(1)$, then the allyl carbons are $CH_2(2)$=$CH(3)$? No — allyl is $CH_2$–$CH$=$CH_2$, so numbering from O outward: $O$ is 1, the attached $CH_2$ is 2, the internal alkene carbon is 3, the terminal alkene carbon would be 4 — but that overcounts, because the $\sigma$ bond breaking is O–$CH_2$, so oxygen and that $CH_2$ are the position-1 pair on each side of the breaking bond, and the allyl chain is numbered independently starting at that same $CH_2$ as position 1. Restart cleanly using the correct method (numbering always starts at the atoms of the breaking bond, not at some other reference point): the breaking bond is O–CH$_2$(allyl). Fragment A = the oxygen side, which continues into the propargyl group: $O$ is position 1, the propargyl $CH_2$ is position 2, the alkyne carbon attached to it is position 3, the terminal alkyne carbon is position 4. Fragment B = the allyl side: the $CH_2$ originally bonded to O is position 1, the internal alkene carbon is position 2, the terminal alkene carbon is position 3. New bond forms between position 3 of the oxygen/propargyl fragment (an alkyne carbon) and position 3 of the allyl fragment (the terminal alkene carbon) — a $[3,3]$-shift, exactly analogous to the ordinary Claisen rearrangement, except that the enol ether oxygen's other substituent is an alkynyl (propargyl) group rather than a vinyl group. The product is an allenyl aldehyde after tautomerization (the initially formed enol, bearing a cumulated-diene/allenic unit from the former alkyne, tautomerizes to an $\alpha,\beta$-unsaturated aldehyde bearing a pendant allene). Ring size $3+3=6$, $N_\pi = 4$, matching the standard six-electron Claisen-type transition state — confirming that despite the extra unsaturation (a triple bond instead of a double bond) on one arm, the order label is unaffected, because the alkyne still contributes exactly one $\pi$ system position between "1" and "3" on that fragment, just as an alkene would; the label counts atoms, not bond multiplicity.

:::warning[The two numbers in [i,j] need not be, and often are not, equal] Students overexposed to the Cope/Claisen $[3,3]$ case sometimes assume every sigmatropic shift is symmetric. It is not: $[1,3]$, $[1,5]$, $[2,3]$, and even higher shifts like $[1,7]$ (relevant to vitamin D biosynthesis, where a hydrogen migrates across a triene system in the ring-opened secosteroid) are all well-precedented, unsymmetrical $[i,j]$ processes. Always determine $i$ and $j$ independently by tracing each fragment separately from the breaking bond outward; never assume symmetry.

::: Two additional scope notes matter for correct classification. First, a sigmatropic rearrangement requires the migrating $\sigma$ bond to be flanked by a $\pi$ system on at least the fragment that is not the migrating terminus itself — a lone sp$^3$–sp$^3$ $\sigma$ bond with no adjacent unsaturation on either side simply cannot undergo a pericyclic sigmatropic shift; there is no conjugated pathway for the electrons to flow through in a single arrow-pushing cycle. Second, when both fragment counts exceed 1 (as in $[3,3]$ or $[2,3]$), the stereochemistry of each fragment (E/Z geometry of alkenes, R/S configuration at stereocenters) becomes translated into the product stereochemistry through the geometry of the transition state (chair or boat), which is the subject taken up in detail once the suprafacial/antarafacial vocabulary is introduced.

The systematic classification of sigmatropic rearrangements, including the $[i,j]$ nomenclature itself, was introduced by Robert Burns Woodward and Roald Hoffmann as part of the broader orbital-symmetry framework they built across a series of papers in the mid-1960s, extending the logic they had already developed for electrocyclic ring closures and cycloadditions. Woodward was already the most celebrated total synthesis chemist of his generation — his syntheses of quinine relay compounds, cholesterol, cortisone, strychnine, reserpine, and later vitamin B$_{12}$ (with Albert Eschenmoser) had established him as the master craftsman of complex molecule assembly built from mechanistic first principles rather than trial-and-error. Hoffmann, a young theoretical chemist trained under William Lipscomb and steeped in extended Hückel molecular orbital calculations, supplied the orbital-symmetry machinery: the notion that the conservation of orbital symmetry through a concerted transition state dictates whether a pericyclic process is thermally or photochemically allowed. Their collaboration began almost by accident: Woodward, puzzling over the stereospecific ring-opening of substituted cyclobutenes observed during his vitamin B$_{12}$ work, recognized a pattern connecting orbital phase and ring geometry, and asked Hoffmann — then a junior faculty member at Cornell — to check the idea against molecular orbital calculations. The resulting Woodward–Hoffmann rules unified electrocyclic reactions, cycloadditions, and sigmatropic shifts under a single conceptual roof: conservation of orbital symmetry along the reaction coordinate. Hoffmann shared the 1981 Nobel Prize in Chemistry with Kenichi Fukui (who had independently developed frontier molecular orbital theory reaching similar conclusions); Woodward, who died in 1979, was ineligible for the prize posthumously but is universally regarded as co-author of the rules that bear both their names.