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How to write down a presentation of a Lie algebra if we know a set of generators in matrix form? For example, for $sl_2$, if we know $e=(0, 1; 0, 0)$, $f=(0, 0; 1, 0)$ , $h=(1, 0; 0, -1)$, how to write a presentation of $sl_2$ from these matrices?

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What you need to compute is the Lie bracket of the generators, in this case this is just the commutator. Because of antisymmetry, you just have to compute one direction: \begin{align*} [e,f]&=ef-fe=\begin{pmatrix}0&1\\0&0\end{pmatrix}\begin{pmatrix}0&0\\1&0\end{pmatrix}-\begin{pmatrix}0&0\\1&0\end{pmatrix}\begin{pmatrix}0&1\\0&0\end{pmatrix}\\ &=\begin{pmatrix}1&0\\0&-1\end{pmatrix}=h\\ [e,h]&=eh-he=\begin{pmatrix}0&1\\0&0\end{pmatrix}\begin{pmatrix}1&0\\0&-1\end{pmatrix}-\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}0&1\\0&0\end{pmatrix}\\ &=-2e \end{align*} I leave the computation for $[f,h]$ for you.

In googling this question I found a nice paper on the issue how to convert the different possibilities of presenting a Lie algebra into each other. It is Cohen, de Graaf, Ronyai: Computations in finite-dimensional Lie algebras, Discrete Mathematics and Theoretical Computer Science, 1997.

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