The maximal dimension of a nilpotent subspace of b-symmetric endomorphisms over a field of characteristic 2 is ν(n−ν), and for b-alternating endomorphisms it is ν(n−ν−1) except when n=2ν+1, where dim SKerQ must be subtracted.
de Seguins Pazzis, Invitation aux formes quadratique s, Calvage & Mounet, Paris, 2011
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The structured Gerstenhaber problem (III)
The maximal dimension of a nilpotent subspace of b-symmetric endomorphisms over a field of characteristic 2 is ν(n−ν), and for b-alternating endomorphisms it is ν(n−ν−1) except when n=2ν+1, where dim SKerQ must be subtracted.