Editing Semisimple Lie algebra (section)

=== Noncompact case ===
Suppose <math>\mathfrak g</math> is not necessarily a compact form (i.e., the signature of the Killing form is not all negative). Suppose, moreover, it has a [[Cartan involution]] <math>\theta</math> and let <math>\mathfrak g = \mathfrak k \oplus \mathfrak p</math> be the eigenspace decomposition of <math>\theta</math>, where <math>\mathfrak k, \mathfrak p</math> are the eigenspaces for 1 and -1, respectively. For example, if <math>\mathfrak g = \mathfrak{sl}_n \mathbb{R}</math> and <math>\theta</math> the negative transpose, then <math>\mathfrak k = \mathfrak{so}(n)</math>.

Let <math>\mathfrak a \subset \mathfrak p</math> be a maximal abelian subspace. Now, <math>\operatorname{ad}(\mathfrak p)</math> consists of symmetric matrices (with respect to a suitable inner product) and thus the operators in <math>\operatorname{ad}(\mathfrak a)</math> are simultaneously diagonalizable, with real eigenvalues. By repeating the arguments for the algebraically closed base field, one obtains the decomposition (called the '''restricted root space decomposition'''):<ref>{{harvnb|Knapp|2002|loc=Ch. V, § 2, Proposition 5.9.}}</ref>
:<math>\mathfrak g = \mathfrak g_0 \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}</math>
where
*the elements in <math>\Phi</math> are called the [[restricted root]]s,
*<math>\theta(\mathfrak{g}_{\alpha}) = \mathfrak{g}_{-\alpha}</math> for any linear functional <math>\alpha</math>; in particular, <math>-\Phi \subset \Phi</math>,
*<math>\mathfrak g_0 = \mathfrak a \oplus Z_{\mathfrak k}(\mathfrak a)</math>.
Moreover, <math>\Phi</math> is a [[root system]] but not necessarily reduced one (i.e., it can happen <math>\alpha, 2\alpha</math> are both roots).