I try to understand an excerpt from the book "Dirac Operators on Representation Theory" by Jing Song Huang and Pavle Pandzic. On page 9 we have following:
A Lie algebra $\frak{g}$ is called semisimple if it is a direct sum of simple ideals, and reductive if it is the direct sum of its center and a semisimple ideal. For example, $\frak{g}l(n, \mathbb{F})$ is reductive: it has a one-dimensional center consisting of scalar matrices, and a direct complement to the center is the simple Lie algebra $\frak{sl}(n, \mathbb{F})$.
A Lie group $G$ is called semisimple if the Lie algebra of $G$ is semisimple. Following Wallach [W] we define a real reductive group or a reductive Lie group as follows. Let $f_1,... , f_m$ be complex polynomials on $M(n,\mathbb{C})$ such that each $f_i$ is real-valued on $M(n,\mathbb{R})$ and such that the set of simultaneous zeros of $f_i$ in $GL(n,\mathbb{C})$ is a subgroup $G_{\mathbb{C}}$ of $GL(n,\mathbb{C})$. Then $G_{\mathbb{C}}$ is called an affine algebraic group defined over $\mathbb{R}$. The subgroup $G_{\mathbb{R}} = G_{\mathbb{C}} \cap GL(n,\mathbb{R})$ is called the group of real points. By a real reductive group or a reductive Lie group we mean a finite covering group $G$ of an open subgroup $G_0$ of $G_{\mathbb{R}}$.
For example, $GL(n, \mathbb{F})$ for $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$ is reductive and every connected semisimple Lie group with finite center is reductive. Thus, we can define a Cartan involution on Lie algebra $\frak{g}$ of a reductive Lie group $G$ by $\theta(X) = −X^* = −\overline{X}^t$.
Obviously $GL(n, \mathbb{F})$ ia reductive because it's Lie-algebra is reductive. Why is every connected semisimple Lie group with finite center reductive as well?
Additionally I not understand why the Cartan involution $\theta$ on the on Lie algebra $\frak{g}$ is well defined. In other words why for $X \in \frak{g}$ the element $−\overline{X}^t$ is as well contained in $\frak{g}$?