# General linear lie algebra

## Homework Statement

Let $\mathfrak{g}$ be the vector subspace in the general linear lie algebra $\mathfrak{gl}_4 \mathbb{C}$ consisting of all block matrices $$A=\begin{bmatrix} X & Z\\ 0 & Y \end{bmatrix}$$ where $X,Y$ are any 2x2 matrices of trace 0 and $Z$ is any 2x2 matrix.

You are given that $\mathfrak{g}$ is a lie subalgebra in $\mathfrak{gl}_4 \mathbb{C}$.

Consider $\mathfrak{g}$ as a lie algebra.

Prove that the radical of $\mathfrak{g}$ consists of all matrices $A$ where $X=Y=0$.

You may use the fact that the lie algebra $\mathfrak{sl}_2 \mathbb{C}$ which consists of all 2x2 matrices of trace 0 is simple.