I'm working out some computations on Lie algebras $L$ of low dimensions (by which I mean $3, 4$ or $5$). For my purposes, it is convenient to choose an orthonormal basis $\{e_1, e_2, \ldots, e_n\}$, for which one has $$[e_i, e_j] = \sum_k \alpha_{ijk}e_k,$$ and then prescribe some nonzero $\alpha_{ijk}$ satisfying antisymmetry and Jacobi. However, I would like to make sure that I'm not doing my computations on the same Lie algebra twice when I decide to pick different constants. Hence my question:
Q: What are some easy-to-compute invariants that one can use to distinguish two Lie algebras when the structure constants $\alpha_{ijk}$ are nice, e.g., are all $\pm 1$ or zero?
As an example of what I'm looking for, if $L$ is a Lie algebra, the dimension of $[L, L]$ is fairly easy to compute if you have the structure constants, although it fails to distinguish quite a few examples.
There are a lot of useful invariants, also if the structure constants are not so "nice". The Killing form is already quite useful. If it is non-degenerate, then the Lie algebra $L$ is semisimple, so that we know that $\dim(L)=3$, since the dimensions $4$ and $5$ are impossible then. If $L$ is nilpotent, then the Killing form is identically zero.
Furthermore there are several invariants for low-dimensional Lie algebras, which are used in the theory of degenerations and contractions (from physics); i.e., the maximal dimension of an abelian subalgebra, the dimension of the derivation algebra, and Kostrikin's covariants $$ c_{ij}(L)=\frac{tr ({\rm ad}(x))^i \cdot tr({\rm ad}(y)^j)}{tr({\rm ad}(x)^î{\rm ad}(y)^j)} $$ For more invariants, and a full discussion for all complex $4$-dimensional Lie algebras, see the paper here.