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?
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?
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 $\lambda,\mu$.