In Lie theory it's possible to compute things very explicit using tensor methods. For example, we can use an explicit matrix for each generator $T^a$ and compute the "action" of this generator on an arbitrary vector $v$ of a given representation using ordinary matrix multiplication
$$ T^av = T_{ij}^a v_j $$ Depending on the representation and the group in question this "vector" can be a matrix and the product can involve complicated additional tensors.
Equivalently it's possible to work in more abstract terms using weights and roots (= the weights of the adjoint representation).
How can I compute the "action" of a generator $T$, represented by a root, on an arbitrary vector $v$ of a given representation, represented by a weight?