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arxiv: 2605.25910 · v2 · pith:ALDFFXREnew · submitted 2026-05-25 · 🧮 math.DG · math.OC

G₂ and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square

Nicklas Day , Boris Doubrov , Igor Zelenko This is my paper

Pith reviewed 2026-06-29 20:12 UTC · model grok-4.3

classification 🧮 math.DG math.OC
keywords G2rank 3 distributionsgrowth vector (3,6,8)symmetry algebrasplit octonionsbracket-generatingmaximal symmetryabnormal extremals
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The pith

A rank-3 distribution on an 8-manifold with growth vector (3,6,8) has 29-dimensional symmetry algebra isomorphic to (g2 ⊕ R) ⋉ W.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a bracket-generating rank-3 distribution on an 8-dimensional manifold whose growth vector is (3,6,8) and whose square is exactly 6-dimensional. The infinitesimal symmetries at each point form a 29-dimensional algebra isomorphic to the semidirect product of (g2 plus the reals) with an adjoint module W of g2. This model achieves maximal symmetry in the family of all bracket-generating rank-3 distributions that have a 6-dimensional square. It also serves as the first counterexample to the conjecture that every such distribution must be of maximal class at a generic point, a property previously verified in lower dimensions.

Core claim

We discover a new realization of the split real form of the exceptional Lie group G2 as the symmetry group of a maximally symmetric rank 3 distribution on a 8-dimensional manifold with the small growth vector (3,6,8). The algebra of infinitesimal symmetries of this distribution at any point is 29-dimensional and isomorphic to (g2 ⊕ R) ⋉ W. Our model is maximally symmetric among all bracket-generating rank 3 distributions with a 6-dimensional square and provides the first counterexample to the conjecture that all such distributions are of maximal class at a generic point. Further analysis shows that all abnormal extremal trajectories have a corank of at least 2. We interpret the model in term

What carries the argument

The rank-3 distribution on the 8-manifold with growth vector (3,6,8) and 6-dimensional square, realized via split-octonions, whose symmetry algebra is the 29-dimensional (g2 ⊕ R) ⋉ W.

If this is right

  • It is maximally symmetric among all bracket-generating rank 3 distributions with a 6-dimensional square.
  • It provides the first counterexample to the conjecture that all bracket-generating rank 3 distributions with a 6-dimensional square are of maximal class at a generic point.
  • All abnormal extremal trajectories of this model have a corank of at least 2.
  • This is the first example with this corank property among bracket-generating distributions with generic small growth vector for a given rank and ambient dimension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction indicates that in higher dimensions maximal symmetry may occur with shorter growth vectors rather than the longest ones.
  • The split-octonion model may connect to other geometric realizations of exceptional Lie groups.
  • Analogous counterexamples to the maximal-class conjecture could appear for distributions of different ranks or in other ambient dimensions.

Load-bearing premise

The explicit construction yields a bracket-generating rank-3 distribution whose symmetry algebra is exactly the 29-dimensional semidirect product (g2 ⊕ R) ⋉ W at every point.

What would settle it

A computation showing the symmetry algebra dimension at some point differs from 29 or is not isomorphic to (g2 ⊕ R) ⋉ W, or discovery of another distribution in the family with symmetry algebra larger than 29 dimensions.

read the original abstract

In 1910, \'{E}lie Cartan famously realized the split real form of the exceptional Lie group $G_2$ as the symmetry group of the maximally symmetric rank 2 distribution on a 5-dimensional manifold with the small growth vector (2,3,5). In this paper, we discover a new appearance of $G_2$ in the geometric theory of distributions, arising from a rank 3 distribution on an 8-dimensional manifold with the growth vector $(3,6,8)$. The algebra of infinitesimal symmetries of this distribution at any point is 29-dimensional and isomorphic to $(\mathfrak{g}_2 \oplus \mathbb{R}) \ltimes W$, where $\mathfrak{g}_2$ is the Lie algebra of $G_2$ and $W$ is an adjoint module of $\mathfrak{g}_2$. Our model possesses three remarkable properties. First, it is maximally symmetric among all bracket-generating rank 3 distributions with a 6-dimensional square (a family that includes both (3,6,8) and (3,6,7,8) distributions). To the best of our knowledge, this is the first example of a family of distributions defined by a set of prescribed small growth vectors in which maximal symmetry is achieved by a member whose growth vector is not the longest. Second, this model provides the first counterexample to the conjecture that all bracket-generating rank 3 distributions with a 6-dimensional square are of maximal class at a generic point (which is known to hold in dimensions 6 and 7). Third, further analysis yields the control-theoretic consequence that all abnormal extremal trajectories of this model originating at any point of the ambient manifold have a corank of at least 2. To our knowledge, this is the first example with this property among bracket-generating distributions with generic small growth vector for a given rank and ambient dimension. We also give an interpretation of our model in terms of split-octonions, more precisely, in terms of a natural algebraic structure on the tangent bundle to split octonions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs an explicit bracket-generating rank-3 distribution on an 8-manifold with growth vector (3,6,8) and 6-dimensional square. It computes that the algebra of infinitesimal symmetries at each point is 29-dimensional and isomorphic to (𝔤₂ ⊕ ℝ) ⋉ W. The model is claimed to be maximally symmetric among all bracket-generating rank-3 distributions with 6-dimensional square, supplies the first counterexample to the conjecture that all such distributions are of maximal class at a generic point, yields that all abnormal extremals have corank at least 2, and admits an interpretation via split-octonions.

Significance. If the explicit local frame, growth-vector verification, and symmetry-algebra computation are correct, the result supplies a new geometric realization of the split real form of G₂, the first instance in which maximal symmetry for a prescribed family of small growth vectors is attained by a distribution whose growth vector is not the longest, and the first counterexample to the maximal-class conjecture in dimension 8. The control-theoretic consequence on abnormal extremals is also novel. These findings would be of interest to the sub-Riemannian geometry and Cartan geometry communities.

minor comments (3)
  1. §1 (Introduction): the statement of the conjecture being disproved would benefit from an explicit citation to the source(s) in which it was formulated for dimensions 6 and 7.
  2. The notation for the semidirect product (𝔤₂ ⊕ ℝ) ⋉ W should be accompanied by a brief description of the module action of 𝔤₂ on W already in the abstract or first paragraph of the introduction.
  3. Figure captions (if any) and the local-frame presentation in the model section should be cross-referenced to the growth-vector computation to make the verification of the 6-dimensional square immediate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance to the sub-Riemannian and Cartan geometry communities, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation rests on an explicit local frame (or Pfaffian system) for the rank-3 distribution together with direct verification of the growth vector (3,6,8), the 6-dimensional square, and the 29-dimensional symmetry algebra via explicit computation of infinitesimal symmetries. These steps are algebraic and differential-geometric, performed on the constructed model itself, without any fitted parameters renamed as predictions, self-definitional relations, or load-bearing self-citations that reduce the central claims to their own inputs. The maximality statement and counterexample status follow from direct comparison within the family of bracket-generating distributions with the given growth constraints, keeping the chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of an explicit bracket-generating distribution with the stated growth vector and square dimension whose symmetry algebra can be computed to be the indicated semidirect product.

axioms (1)
  • domain assumption Existence of a bracket-generating rank-3 distribution on an 8-manifold with growth vector (3,6,8) and 6-dimensional square whose symmetry algebra is (g2 ⊕ R) ⋉ W at every point.
    The maximality and counterexample statements presuppose that such a distribution has been constructed and its symmetries verified.

pith-pipeline@v0.9.1-grok · 5928 in / 1335 out tokens · 42154 ms · 2026-06-29T20:12:14.961106+00:00 · methodology

discussion (0)

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