Strictly upper triangular Lie algebra n3(C)

The strictly upper triangular complex matrix Lie algebra $\mathfrak n_{3}(\mathbb C)$ consists of matrices $X=(x_{ij})\in M_{3}(\mathbb C)$ with $x_{ij}=0\text{ whenever }i\geq j$ , using the commutator bracket.

Basis and nilpotency

The basis $E_{ij}\ (i<j)$ has brackets $[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{li}E_{kj}$ . Each additional commutator moves entries farther above the diagonal. In size three the only nonzero basic bracket is $[E_{12},E_{23}]=E_{13}$ , giving the three-dimensional Heisenberg algebra.