Lemma 3

Let A be a central simple K-algebra of finite degree, and let L be an extension of K of finite degree that is a splitting field for A. Let V be a simple $A_{(L)}$-module, so that the natural morphism $\varphi : A_{(L)}\rightarrow$ End$_L(V)$ is an isomorphism. Let C be the ring End$_A(V)$. Then C is similar to $A^o$, and the image of $L\otimes 1\subset A_{(L)}$ is a maximal commutative subalgebra of C.

We identify A with a subring of $A_{(L)}$. We view V as a K-vector space. The ring C is the commutant of $\varphi (A)$ in End$_K(V)$. It is a central simple K-algebra of finite degree, and the homomorphism $a\otimes c\mapsto ac$ from $A\otimes_KC$ to End$_K(V)$ is an isomorphism (VIII, p. 260, Theorem 6, a)). Consequently, the K-algebras A and $C^o$ are similar (VIII, p. 279, Proposition 3).

Let $L_V$ be the ring of homotheties of the L-vector space V; it is the commutant of End$_L(V)$ in End$_K(V)$ (VIII, p. 82, Corollary 1). Now, the K-algebra End$_L(V)$ is generated by $\varphi (A)$ and $L_V$; therefore, in End$_K$(V), we have

$L_V=$ End$_L(V)'=\varphi (A)'\cap L'_V= C\cap L'_V$,

where for any subset B of End$_K$(V), the commutant of B in End$_K(V)$ is denoted by $B'$. So $L_V$ is a maximal commutative subalgebra of C (VIII, p. 261, Lemma 3), and therefore also of $C^o$. The mapping $\lambda \mapsto \lambda_V$ is a K-algebra isomorphism from L to $L_V$. This proves the lemma.