Suppose we have a Lie algebra characterized not by the Lie bracket, but by the exterior derivative, that is, if $\{e^1,...,e^n\}$ is a dual basis, then we have

$$\mathrm{d}e^i=\sum_{j<k}a_{ijk}e^j\wedge e^k$$

for some coefficients $a_{ijk}$.

In this setting, how may be checked (perhaps computationally?) that our algebra is nilpotent? I wonder whether there is a somewhat nice algorithm involving the coefficients.

Thank you for your attention and answers.

Suppose we have a Lie algebra with structure constants

$$\mathrm{d}e^i=\sum_{j<k}a_{ijk}e^j\wedge e^k$$

for some coefficients $a_{ijk}$.

In this setting, how may be checked (perhaps computationally?) that our algebra is nilpotent? I wonder whether there is a somewhat nice algorithm involving the coefficients.

Thank you for your attention and answers.