Dniester Notebook 1.55: exhibit an explicit finite basis of identities for the Cayley-Dickson (split-octonion) algebra

The *Cayley–Dickson algebra* here is the 8-dimensional split-octonion (Zorn vector-matrix) algebra $\mathbb{O}$ over a field; a *basis of identities* is a set of polynomial identities of $\mathbb{O}$ from which all its identities follow. Zhevlakov's Dniester Notebook Problem 1.55 asks to find such a basis (a) over a finite field, (b) over a field of characteristic 0, and (c) over an infinite field of prime characteristic. Part (a) is solved (Isaev, 1984). For part (b), Iltyakov (1991) proved that a finite basis EXISTS (the identities of $\mathbb{O}$ are finitely based in characteristic 0) — but this is a pure existence theorem: no explicit basis has been written down and proved complete. The open content of (b) is therefore to exhibit one; the concrete target is the Bremner–Madariaga–Peresi conjecture that the known octonion identities of degree $\le 6$ generate all identities of $\mathbb{O}$ in characteristic 0 (they computed all multilinear identities through degree 7 and found none new in degree 7). Part (c) is open even at the level of finite-basis existence. Exhibit and prove an explicit finite basis of identities for $\mathbb{O}$ in characteristic 0 — in particular prove or refute the degree-$\le 6$ conjecture — and/or settle whether the identities of $\mathbb{O}$ over an infinite field of prime characteristic are finitely based.

Acceptance. FULLY RESOLVES (b): an explicit finite set of identities of the split-octonion algebra in characteristic 0 together with a proof that it generates all identities of the algebra — in particular a proof or refutation of the Bremner–Madariaga–Peresi conjecture that the degree-<=6 identities suffice — with re-runnable computer-algebra code verifying the relevant multilinear identity spaces (as symmetric-group modules). FULLY RESOLVES (c): a proof that the identities of the split-octonion algebra over an infinite field of prime characteristic are, or are not, finitely based. ADVANCES: a certified extension of the multilinear-identity computation to degree 8 or beyond (S_n-module decomposition with rational and/or modular dual certificates), showing whether new generators appear; or a fully proved basis for a distinguished subspace of identities (e.g. the skew-symmetric ones). All computational claims must ship exact, re-runnable verification code.