# Lie Algebra over Quaternions

We define the skew-symmetric hermitian inner product $$\phi(x,y)=\bar{x}^tjy$$ over $\mathbb{H}^2$ and are asked to calulate the Lie algebra of the group $G\le GL(2,\mathbb{H})$ of automorphisms of $\phi$.

My solution is the following:

We need to find the Lie algebra of $\{X\in GL(2,\mathbb{H})|\bar{X}^tjX=X\}$ so if we take $a(t)\in G$ with $a(0)=1$ and differentiate then we get the condition that:

$\bar{X}^t=jXj$

So the Lie algebra $\mathfrak{g}=\{X\in GL(2,\mathbb{H})|\bar{X}^t=jXj\}$.

I am then asked to calculate the real dimension of this which is where I am having trouble.

If we take $X=\left ( \begin{array}{cc} a & b \\ c & d \end{array} \right )=\left ( \begin{array}{cc} j\bar{a}j & j\bar{c}j \\ j\bar{b}j & j\bar{d}j \end{array} \right )$.

If I now want to calculate the real dimension of this what do I do?

Thanks for any help.

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