There is a notion of “non-commutative Lie algebra” called *Leibniz algebra*, which is characterized by the following property. The bracketing [-, *z*] is a derivation for the bracket operation, that is, it satisfies the Leibniz identity:
$$
\left[ {\left[ {x,y} \right],z} \right] = \left[ {\left[ {x,z} \right],y} \right] + \left[ {x,\left[ {y,z} \right]} \right]
$$
cf. [LI]. When it happens that the bracket is skew-symmetric, we get a Lie algebra since the Leibniz identity becomes equivalent to the Jacobi identity.