Strictly upper triangular Lie algebra nn(R)

For an integer $n\geq1$ , the strictly upper triangular real matrix Lie algebra $\mathfrak n_{n}(\mathbb R)$ consists of matrices $X=(x_{ij})\in M_{n}(\mathbb R)$ 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. For $n\geq2$ the nilpotency class is exactly $n-1$ : successive commutators of $E_{12},E_{23},\ldots,E_{n-1,n}$ reach $E_{1n}$ .