Example: strictly upper triangular matrices

Let $\mathfrak n_n\subset \mathfrak{gl}_n(\mathbb R)$ be the vector space of strictly upper triangular matrices (zeros on and below the diagonal), with Lie bracket the commutator $[A,B]=AB-BA$.

This is a standard example of a nilpotent Lie algebra .

Concrete bracket computation in the matrix-unit basis

For $i<j$, let $E_{ij}$ be the matrix unit. A direct multiplication gives

so

In particular, if $i<j$ and $k<l$, then $[E_{ij},E_{kl}]$ is either $0$ or another strictly upper triangular matrix unit.

Lower central series (explicit nilpotency mechanism)

Define the “height” of $E_{ij}$ as $j-i\ge 1$. From the formula above, any nonzero commutator of strictly upper triangular matrices increases height: the product $E_{ij}E_{jk}=E_{ik}$ has height $(k-i)=(j-i)+(k-j)$.

Consequently, iterated commutators eventually vanish: the lower central series

satisfies $\gamma_{n}=0$. Thus $\mathfrak n_n$ is nilpotent of class at most $n-1$, hence solvable (compare nilpotent implies solvable and derived series ).

Small case $n=3$ (fully explicit)

A general element is

Using $[E_{12},E_{23}]=E_{13}$ and other brackets $0$, we get

which matches the Heisenberg algebra pattern.