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Algebra Comput."],"published-print":{"date-parts":[[2026,5]]},"abstract":"<jats:p>The structure of maximal nilpotent linear subspaces of [Formula: see text], and more generally of [Formula: see text] for any field F with at least three elements, is now well established. In this work, we extend the analysis to dimension five. The main result of this paper consists of the description, up to similarity, of all two-dimensional nilpotent linear subspaces of [Formula: see text]. We expect that these results will allow us to describe every maximal nilpotent linear subspace of [Formula: see text]. We think that this classification is central to the solution of the classical problem of Albert about finite-dimensional power-associative nilalgebras with dimension \u226510. 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