reductive Lie group

《Lie Groups Beyond an Introduction》 · 第 446 页 · Chapter VII，Section 2「Reductive Lie Groups」

A reductive Lie group is actually a 4-tuple $(G, K, \theta, B)$ consisting of a Lie group $G$, a compact subgroup $K$ of $G$, a Lie algebra involution $\theta$ of the Lie algebra $\mathfrak g_0$ of $G$, and a nondegenerate, $\operatorname{Ad}(G)$ invariant, $\theta$ invariant, bilinear form $B$ on $\mathfrak g_0$ such that (i) $\mathfrak g_0$ is a reductive Lie algebra, (ii) the decomposition of $\mathfrak g_0$ into $+1$ and $-1$ eigenspaces under $\theta$ is $\mathfrak g_0 = \mathfrak k_0 \oplus \mathfrak p_0$, where $\mathfrak k_0$ is the Lie algebra of $K$, (iii) $\mathfrak k_0$ and $\mathfrak p_0$ are orthogonal under $B$, and $B$ is positive definite on $\mathfrak p_0$ and negative definite on $\mathfrak k_0$, (iv) multiplication, as a map from $K \times \exp \mathfrak p_0$ into $G$, is a diffeomorphism onto, and (v) every automorphism $\operatorname{Ad}(g)$ of $\mathfrak g = (\mathfrak g_0)^\mathbb C$ is inner for $g \in G$, i.e., is given by some $x$ in $\operatorname{Int} \mathfrak g$.