Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying
$$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$
Let $x, y \in A$ such that $xy = yx$ and such that $V = \mathbb{R}x + \mathbb{R}y$ is of dimension 2 over $\mathbb{R}$. Show that
$$\forall u, v \in V \quad \|u+v\|^2 + \|u-v\|^2 \geq 4\|u\| \cdot \|v\|$$
and that the restriction of $\|\cdot\|$ to $V$ comes from an inner product on $V$.