Lemma 4

Let A be a central simple algebra of finite degree over K, distinct from K. There exists an étale (V, §6, No. 3, p. 28, Definition 1) subalgebra of A distinct from K.

By Wedderburn’s theorem (VIII, p. 120, Theorem 1), we may assume that A is of the form $\mathbf{M}_n$(D), where $n$ is a strictly positive integer and D a field with center K.

Suppose $n >1$. The algebra of diagonal matrices with entries in K is an étale subalgebra of A distinct from K.

Suppose $n= 1$. Let $p$ be the characteristic exponent of D. By Lemma 1 of VIII, p. 230, there exists an element $a$ of D such that $a^{p^m}$ does not belong to K for any natural number $m$. So for $m$ sufficiently large, the element $x=a^{p^m}$ is separable over K (V, §7, No. 7, p. 44, Proposition 13), but it does not belong to K. The subalgebra $K(x)$ of A is a separable extension of the field K of finite degree, hence an étale subalgebra over K; it is distinct from K.