Commutator on Algebra is Alternating Bilinear Mapping

Theorem
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring.

Then the commutator on $\left({A_R, \oplus}\right)$ is an alternating bilinear mapping:


 * $\forall a, b \in A_R: \left[{a, b}\right] = -\left[{b, a}\right]$