Let $D$ be a division algebra and $n\in \mathbb{N}$. If $D$ is a field, then it is well-known that the diagonal-matrices form a Cartan subalgebra of $gl(n,D)$. Is there a complete description of all Cartan subalgebras?
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It can be proven that for an arbitrary associative algebra A and a Cartan subalgebra C of the Lie algebra associated to A the set of diagonal matrices over C is a Cartan subalgebra of $gl(n,A)$. – Sven Wirsing Sep 10 '16 at 08:07
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partial answers given here: http://mathoverflow.net/questions/249661/cartan-subalgebras-of-matrix-algebras-over-fields-and-division-algebras/249866#249866 – Sven Wirsing Sep 27 '16 at 19:12