Lie Algebra of $SL_n(\mathbb H)$ The Lie algebra of $SL_n(\mathbb C)$ are the matrices where the trace is $0$. But what is the Lie algebra of $SL_n(\mathbb H)$ where $\mathbb H$ is the quaternions?

The group $\mathrm{SL}(n, \mathbb{H})$ is defined here:

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*Wikipedia, GL(n,H) and SL(n,H).

It works like this:
The algebra of quaternions is a subalgebra of the $2 \times 2$ complex matrices in a standard way. This lets us regard $n \times n$ quaternionic matrices as certain special $2n \times 2n$ complex matrices.  This in turn lets us define a complex-valued determinant of an $n \times n$ quaternionic matrix, called the Study determinant, and also a complex-valued trace.
The $n \times n$ quaternionic matrices for which the Study determinant is 1 form the group $\mathrm{SL}(n,\mathbb{H})$.   The Lie algebra of $\mathrm{SL}(n,\mathbb{H})$ then consists of $n \times n$ quaternionic matrices for which the complex-valued trace is 0.
The Study determinant is the square of the Dieudonné determinant, as shown here:

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*Helmer Aslaksen, Quaternionic determinants, The Mathematical Intelligencer 18 (1996), 57–65.

But the Dieudonné determinant is nonnegative, so in fact the Study determinant also takes only nonnegative real values, and $\mathrm{SL}(n,\mathbb{H})$ is also the group of quaternionic $n \times n$ matrices with Dieudonné determinant 1.
