# Derived lie algebra

1. Sep 23, 2011

### Ted123

1. The problem statement, all variables and given/known data

Find the derived lie algebra of $\mathfrak{so}_3 \mathbb{C}$, the 3x3 antisymmetric matrices with entries in $\mathbb{C}$ with Lie bracket the matrix commutator $[X,Y]=XY-YX$ for any $X,Y\in \mathfrak{so}_3 \mathbb{C}$.

3. The attempt at a solution

Since $\mathfrak{so}_3 \mathbb{C}$ is simple the derived lie algebra of $\mathfrak{so}_3 \mathbb{C}$ (an ideal) must equal $\mathfrak{so}_3 \mathbb{C}$. Is this right? If so, how do I justify this?

Last edited: Sep 23, 2011