Semisimple Lie group

Remarks

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic $0$. The following are equivalent:

1. the only solvable ideal of $\mathfrak{g}$ is $0$;
2. $\mathfrak{g}$ is a direct sum of simple ideals;
3. the Killing form $B(x, y) = \tr(\operatorname{ad} x \operatorname{ad} y)$ is nondegenerate (Cartan's criterion).

Example: $G = SO(4)$. Its Lie algebra $\mathfrak{so}(4)$ is isomorphic to $\mathfrak{so}(3) \oplus \mathfrak{so}(3)$, a direct sum of two simple ideals, so $G$ is semisimple. It is not simple, since each summand is a proper nonzero ideal.

Non-example: $G = GL_n(\R)^+$, the invertible real $n \times n$ matrices with positive determinant, for $n \ge 1$. Its Lie algebra $\mathfrak{gl}_n(\R)$ contains the ideal $\R I_n$ of scalar matrices, which is abelian, hence solvable, and nonzero. So the condition fails.

Some authors do not require $G$ to be connected and ask only that $\mathfrak{g}$ be semisimple; some also ask for a finite centre.