Proposition 4

Let A be a central simple algebra of finite degree over K. Let L be a subalgebra of A, and let $L'$ be the commutant of L in A.

a) If L is maximal among the semisimple commutative subalgebras of A, then we have $L = L'$, and L is a maximal commutative subalgebra of A.

b) If L is maximal among the étale subalgebras of A, then we have $L = L'$, and L is a maximal commutative subalgebra of A.

We know that the relation $L = L'$ means that L is a maximal commutative subalgebra of A (VIII, p. 261, Lemma 3, a)). Suppose that L is semisimple, commutative, and distinct from $L'$. By Theorem 5 of VIII, p. 259$, L'$ is semisimple, and L is the commutant of $L'$ and therefore the center of $L'$; consequently, $L'$ is not commutative. It suffices to prove that there exists a semisimple commutative subalgebra M of A that is distinct from L and contains L and that is étale if L is étale.

By the structure theorem for semisimple rings (VIII, p. 135, Theorem 1), there exist simple rings $B_1, . . . ,B_r$ and an isomorphism $\varphi$ from $L'$ to $B_1\times \cdots \times B_r$. For $1\leqslant i\leqslant r$, denote the center of $B_i$ by $E_i$; we then have $\varphi (L) = E_1\times \cdots \times E_r$. Since $L'$ is not commutative, we may assume that $B_1$, for example, is not commutative; we therefore have $B_1\not= E_1$, and, by Lemma 4, there exists a subalgebra $M_1$ of $B_1$ that is commutative, distinct from $E_1$, and étale over $E_1$. Set $M =\varphi^{-1}(M_1\times E_2\times \cdots \times E_r)$; it is a semisimple commutative subalgebra of A containing L and distinct from L. Suppose that L is étale over K; let us prove that M is étale. The extensions $E_i$ of K are separable (V, §6, No. 4, p. 30, Proposition 3). Moreover, since the $E_1$-algebra $M_1$ and the K-algebra $E_1$ are étale, the K-algebra $M_1$ is also étale (V, §6, No. 5, p. 32, Corollary 2). Thus the K-algebra $M_1\times E_2\times \cdots \times E_r$ is étale, and therefore so is M.

Let A be a central simple algebra of finite degree over K. A subalgebra of A that is maximal among the semisimple commutative subalgebras of A is called a maximal semisimple commutative subalgebra of A. By Proposition 4, the term “maximal” refers to the property of being commutative or of being semisimple and commutative. A subalgebra of A that is maximal among the étale subalgebras of A is called a maximal étale subalgebra of A.