Corollary

Let A be a central simple algebra of finite degree over K, and let L be a semisimple commutative K-algebra such that [A : K] is equal to $[L : K]^2$. Let $f$ and $g$ be injective homomorphisms from L to A. There exists an inner automorphism $\theta$ of A such that $g=\theta \circ f$.

Set $n= [L : K]$. Viewed as a left module over the subring $f$(L), A is free of dimension $n$; this follows from the equivalence of properties (ii) and (iii) of Proposition 3. Since $f$ is an isomorphism from L to $f$(L), the left L-module $A^f$ (whose law of action is given by $(x, a)\mapsto f(x)a$) is free of dimension $n$. The same holds for $A^g$, which is therefore isomorphic to $A^f$. We conclude using the equivalence of properties (i) and (ii) of Proposition 1 (VIII, p. 257).

Suppose that A is a central simple algebra of finite degree over K. There can exist maximal commutative subalgebras L of A that are not semisimple and for which $[A : K]\not= [L : K]^2$ (VIII, p. 270, Exercise 5).