Let $\mathfrak{g}$ be a Lie algebra. Then the nilradical $\mathfrak{n}$ of $\mathfrak{g}$ is defined to be the intersection of the kernels of all the irreducible representations of $\mathfrak{g}$. Equivalently, $\mathfrak{n}=[\mathfrak{g},\mathfrak{g}]\cap\mathrm{rad}\,\mathfrak{g}$, the intersection of the derived ideal and radical of $\mathfrak{g}$.