 An alternative definition can be given to make the bracket skew-symmetric as

This no longer satisfies the Jacobi-identity axiom above.It instead fulfills a homotopic Jacobi-identity.where "T" is

The Leibniz rule and the invariance of the scalar product become modified by the relation formula_17 and the violation of skew-symmetry gets replaced by the axiom 

The skew-symmetric bracket together with the derivation "D" and the Jacobiator "T" form a strongly homotopic Lie algebra.