Commutator on Algebra is Alternating Bilinear Mapping

Theorem
Let $\struct {A_R, \oplus}$ be an algebra over a ring.

Then the commutator on $\struct {A_R, \oplus}$ is an alternating bilinear mapping:


 * $\forall a, b \in A_R: \sqbrk {a, b} = -\struct {b, a}$