Proposition 1

Let $\mathfrak{g}$ be a (real) Lie algebra. The following conditions are equivalent:

(i) $\mathfrak{g}$ is isomorphic to the Lie algebra of a compact Lie group.

(ii) The group Int($\mathfrak{g}$) (Chap. III, §6, no. 2, Def. 2) is compact.

(iii) $\mathfrak{g}$ has an invariant bilinear form (Chap. I, §3, no. 6) that is symmetric, positive and separating.

(iv) $\mathfrak{g}$ is reductive (Chap. I, §6, no. 4, Def. 4); for all $x\in \mathfrak{g}$, the endomorphism ad $x$ is semi-simple, with purely imaginary eigenvalues.

(v) $\mathfrak{g}$ is reductive and its Killing form B is negative.

(i) $=\Rightarrow$ (ii): if $\mathfrak{g}$ is the Lie algebra of a compact Lie group G, the group Int($\mathfrak{g}$) is separating and isomorphic to a quotient of the compact group $G_0$ (Chap. III, §6, no. 4, Cor. 4), hence is compact.

(ii) $=\Rightarrow$ (iii): if the group Int($\mathfrak{g}$) is compact, there exists a symmetric bilinear form on $\mathfrak{g}$ that is positive, separating and invariant under Int($\mathfrak{g}$) (no. 1), hence also invariant under the adjoint representation of $\mathfrak{g}$.

(iii) $=\Rightarrow$ (iv): if (iii) is satisfied, the adjoint representation of $\mathfrak{g}$ is semi-simple (no. 1), hence $\mathfrak{g}$ is reductive; moreover, the endomorphisms ad $x$, for $x\in \mathfrak{g}$, have the indicated properties (no. 1).

(iv) $=\Rightarrow$ (v): for all $x\in \mathfrak{g}, B(x, x) =$ Tr((ad $x$)$^2$); consequently, $B(x, x)$ is the sum of the squares of the eigenvalues of ad $x$, and hence is negative if these are purely imaginary.

(v) $=\Rightarrow$ (i): assume that $\mathfrak{g}$ is reductive, hence the product of a commutative subalgebra $\mathfrak{c}$ and a semi-simple subalgebra $\mathfrak{s}$ (Chap. I, §6, no. 4, Prop. 5). The Killing form of $\mathfrak{s}$ is the restriction of the form B to $\mathfrak{s}$, hence is negative and separating if B is negative. The subgroup Int($\mathfrak{s}$) of $\mathbf{G}\mathbf{L}(\mathfrak{s})$ is closed (it is the identity component of Aut($\mathfrak{s}$), Chap. III, §10, no. 2, Cor. 2) and leaves the separating positive form $-B$ invariant; thus, it is compact, and $\mathfrak{s}$ is isomorphic to the Lie algebra of the compact Lie group Int($\mathfrak{s}$). Further, since $\mathfrak{c}$ is commutative, it is isomorphic to the Lie algebra of a torus T. Thus $\mathfrak{g}$ is isomorphic to the Lie algebra of the compact Lie group Int($\mathfrak{s}$)$\times T$.