Strictly upper triangular Lie algebra n2(R)

The strictly upper triangular real matrix Lie algebra $\mathfrak n_{2}(\mathbb R)$ consists of matrices $X=(x_{ij})\in M_{2}(\mathbb R)$ 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. In size two this is the one-dimensional abelian algebra spanned by $E_{12}$ .