Simple Lie group

Remarks

Example: $G = SU(2)$, with Lie algebra $\mathfrak{su}(2) \cong \R^3$ and the cross product as bracket. The bracket is not identically zero, so clause 1 holds. If an ideal $I$ contains $x \ne 0$, it contains $x \times y$ for all $y$, and these vectors fill the plane $x^\perp$. The cross product of two independent vectors of that plane is a nonzero multiple of $x$, so $I = \R^3$, and clause 2 holds.

Non-example: $G = U(n)$ with $n \ge 2$. Its Lie algebra $\mathfrak{u}(n)$ of skew-Hermitian matrices contains the ideal $i\R\, I_n$ of scalar matrices, which is neither $0$ nor $\mathfrak{u}(n)$, so clause 2 fails. For $n = 1$ the group $U(1)$ is abelian, so clause 1 fails.

For connected $G$, the connected normal Lie subgroups of $G$ correspond to the ideals of $\mathfrak{g}$. So $G$ is simple exactly when it is not abelian and has no connected normal Lie subgroups other than $\set{e}$ and $G$. A simple Lie group may have a nontrivial discrete centre, so it need not be a simple group in the sense of abstract group theory. Some authors also require the centre to be trivial or finite.