Geodesic
On Reductive Lie Admissible Algebras
Canadian journal of mathematics, Tome 23 (1971) no. 2, pp. 325-331

Voir la notice de l'article provenant de la source Cambridge University Press

A Lie admissible algebra is a non-associative algebra A such that A − is a Lie algebra where A − denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A− : Y → [X, Y] and H = {L− (X): X ∊ A −}; then, since A − is a Lie algebra, we see that H is contained in the derivation algebra of A − and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L −(V) ∊ g, then and note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A − is Lie. 
Sagle, Arthur A. On Reductive Lie Admissible Algebras. Canadian journal of mathematics, Tome 23 (1971) no. 2, pp. 325-331. doi: 10.4153/CJM-1971-032-9
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