Cartan subalgebras of matrix algebras over fields and division algebras 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?
It is well-known that for algebraic closed field $D$ all Cartan subalgebras are conjugated. But what is valid for an arbitrary (skew)-field $D$?
(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 $gl(n,A)$.)
 A: Let $D$ be a field of characteristic $0$. Then $L = \mathfrak{gl}_n(D)$ is a split reductive Lie algebra over $D$ with centre $Z(L) = D \cdot E_n$ (where $E_n$ is the $m\times n$ identity matrix), all Cartan subalgebras (= CSAs) are of the form $H= D \cdot E_n \oplus H'$ where $H'$ is a CSA of $L'=\mathfrak{sl}_(D)$. Two CSAs of $L$ are conjugate if and only if their $L'$-parts are conjugate. In general one knows (e.g. [Bourbaki, Lie VIII, 3.3] that all split CSAs are conjugate. But $L$ may have non-split CSAs, see the example on page 108 in Jacobson's book on Lie algebras. The sitation is well-understood for $D= {\mathbb R}$, due to a 1955 paper by Kostant (Proc. Nat. Acad. Sci. U. S. A. 41 (1955), 967–970): There are finitely many conjugacy classes, and one knows representatives of each class. I do not know any reference for other fields, like for example $D= {\mathbb Q}$. 
Let now $D$ be a skew-field. Denote by $K$ its centre, which is a field, and assume that $K$ has characteristic $0$ and that $\dim_K D < \infty$. Then $L$ is a finite-dimensional reductive Lie algebra over $K$, and the CSAs of $L$ all have the same dimension, namely the dimension of the CSAs of $\bar L = L \otimes_K \bar K$, where $\bar K$ is the algebraic closure of $K$. The CSAs of $L$ are all abelian and the elements of a CSA act by semisimple (but not necessrily diagonalizable) endomorphisms in any finite-dimensional semisimple representations of $L$ ([Bourbaki, Lie VII, 2.4]). 
The Lie algebra $L$ fits in the setting of Seligman's 1976 book on "Rational Methods in Lie algebras". If you are willing to replace CSAs by split maximal toral subalgebras, then conjugacy for them is proven there (Ch. I, section 3, Theorem 2]).     
