The dimension of a $\mathbb{R}$-algebra

Question

Let $D$ be an irreducible $\mathbb{R}$-algebra in $M_{n}(\mathbb{C})$ that it implies $D$ has independent vectors as $\{A_{1},...,A_{r^{2}}\}$. Let $D$ is isomorphic to $\mathbb{H}$ (quaternions) that it impiles the dimension of $D$ is $4$ as a $\mathbb{R}$-algebra. Then we want to show $r=2$. To this end, it is enough to show that the dimension of $D$ is $r^{2}$. How?

Answer 1

You are presumably meant to assume that $\{A_{1},...,A_{r^{2}}\}$ is a basis for $D$, not just a set of linearly independent elements. If all you know is that they are linearly independent, then all you can conclude is that $r\leq 2$.