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The author continues to discuss this problem: given a nonzero nilpotent finite-dimensional associative algebra N over the perfect field k, describe the set of unital associative /r-algebras A satisfying the equation rad,4=JV, together with the "nowhere triviality" condition Ann^ N^N. In this paper the Lie homomor- phism ô : SLie-^Der*; TV induced by bracketing (where A has Wedder- burn decomposition as semidirect sum S+N) is studied as follows: (i) the kernel and image of ô are computed ; (ii) conditioning the derivation algebra Derfc TV conditions the semisimple S; (iii) for instance, Derfc TV solvable implies that S is a direct sum of fields ; (iv) those tori in Derfc N of the form OS are characterized in terms of their 0-weightspace in TV.

Radical embedding, genus, and toroidal derivations of nilpotent associative algebras