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Lemma 59.61.2. Let $A$ be a finite central simple algebra over $K$. Then

\[ \begin{matrix} A \otimes _ K A^{opp} & \longrightarrow & \text{End}_ K(A) \\ \ a \otimes a' & \longmapsto & (x \mapsto a x a') \end{matrix} \]

is an isomorphism of algebras over $K$.

Proof. See Brauer Groups, Lemma 11.4.10. $\square$

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