Lie algebra: Efficient way of finding commutation relations

Question

Is there an efficient way of finding commutation relations for a Lie algebra?

For $\mathfrak{su}(2)$ with the Pauli matrices multiplied by $-\frac i2$ we get only three non-trivial commutation relations, but for larger Lie algebras there can be many, many times more.

How does one efficiently find them all? Perhaps they fall into classes of some sort?

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I.e. The basis of the three-dimensional Lie algebra $\mathfrak{sl}(2,\Bbb F)$ is given by:

$$e=\begin{bmatrix}0&1\\0&0\end{bmatrix},\quad h=\begin{bmatrix}1&0\\0&-1\end{bmatrix},\quad f=\begin{bmatrix}0&0\\1&0\end{bmatrix}$$

The corresponding nontrivial commutation relations are:

$$[h,e]=2e,\quad [h,f]=-2f,\quad [e,f]=h$$

Is there a faster way of finding these, than computing the bracket for basis elements and seeing if they are related to another basis element?

Answer 1

In general, given a subalgebra $\mathfrak{h}$ of $\mathbb{gl}_n(K)$ consisiting of matrices, one needs to compute all commutators of a basis $f_1,\ldots ,f_m$ of the vector space $\mathfrak{h}$. Using the canonical basis $E_{ij}$ of $\mathbb{gl}_n(K)$, there are formulas available for the commutators $[E_{ij},E_{kl}]$, i.e., $$ [E_{jk},E_{lm}]=\delta_{kl}E_{jm}-\delta_{jm}E_{lk}. $$ For example, in the case of $\mathbb{sl}_2(K)$ we can choose a basis $e=E_{12}$, $f=E_{21}$ and $h=E_{11}-E_{22}$ and apply the formula. Then we obtain the commutation relations without further computation. Applying more theory we can almost guess the commutator relations, e.g., since $ad(h)$ acts diagonally, we already know that $[h,e]=\lambda e$, $[h,f]=\mu f$ for some scalars $\lambda,\mu$.