Equivalence of Definitions for Alternating Bilinear Mapping on Ring of Characteristic Not 2

Theorem
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$ with the property that $\operatorname{Char}\left({R}\right) \neq 2$.

Then the following definitions for alternating bilinear maps are equivalent:
 * $\oplus$ is an alternating bilinear map for all $a \in A_R$, $a \oplus a = 0$.
 * $\oplus$ is an alternating bilinear map for all $a, b \in A_R$, $a \oplus b + b \oplus a = 0$.

Proof
Let $\oplus$ be a bilinear map with the property that for all $a \in A_R$, $a \oplus a = 0$. Then for all $u, v \in A_R$:

Let $\oplus$ be a bilinear map with the property that for all $a, b \in A_R$, $a \oplus b + b \oplus a = 0$.

Let $u \in A_R$.

Then:

Since the characteristic of $R$ is not $2$, thus:
 * $u \oplus u = 0$