Lower central series of a Lie algebra ($({\gamma }_i(\mathfrak{g}))$)

The lower central series is defined by ${\gamma }_1(\mathfrak{g})=\mathfrak{g}$ and ${\gamma }_{i+1}(\mathfrak{g})=[\mathfrak{g},{\gamma }_i(\mathfrak{g})]$.

A Lie algebra is nilpotent when its lower central series eventually becomes zero.

A finite-dimensional Lie algebra is nilpotent if and only if every adjoint map ${\mathrm{ad}}_x$ is nilpotent. A Lie algebra of nilpotent endomorphisms can be simultaneously represented by strictly upper triangular matrices.