Lemma 1

Let A be a K-algebra of finite degree and D a field of finite degree over K. We have [A] = [D] in $\mathscr{M}_K$ if and only if there exists an integer $n\geqslant 1$ such that A is isomorphic to $\mathbf{M}_n(K)$.

By definition, we have [A] = [D] if and only if there exists an invertible $(A$, D)-bimodule, that is (VIII, p. 101, Theorem 1), a right vector space V of nonzero finite dimension over D endowed with an isomorphism of K-algebras $A\rightarrow$ End$_D(V)$. Lemma 1 follows.