Why are lie algebra of upper-triangular $nxn$ matrices not nilpotent Lie algebra

Question

Is there an easy proof (without Engel's theorem) of the fact that lie algebra of upper-triangular $n\times n$ matrices (of the field $\mathbb{R}$) are not nilpotent Lie algebra?

Answer 1

The Lie algebra $\mathfrak{t}_n(K)$ of upper-triangular matrices cannot be nilpotent, because it has a non-nilpotent solvable Lie subalgebra of dimension $2$, e.g., generated by $E_{12}$ and $E_{11}-E_{22}$. Hence it cannot be nilpotent.