Editing Semisimple Lie algebra (section)

=== Compact case ===
Suppose <math>\mathfrak g</math> is a compact form and <math>\mathfrak h \subset \mathfrak g</math> a maximal abelian subspace. One can show (for example, from the fact <math>\mathfrak g</math> is the Lie algebra of a compact Lie group) that <math>\operatorname{ad}(\mathfrak h)</math> consists of skew-Hermitian matrices, diagonalizable over <math>\mathbb{C}</math> with imaginary eigenvalues. Hence, <math>\mathfrak h^{\mathbb{C}}</math> is a [[Cartan subalgebra]] of <math>\mathfrak{g}^{\mathbb{C}}</math> and there results in the root space decomposition (cf. [[#Structure]])
:<math>\mathfrak{g}^{\mathbb{C}} = \mathfrak{h}^{\mathbb{C}} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}</math>
where each <math>\alpha \in \Phi</math> is real-valued on <math>i \mathfrak{h}</math>; thus, can be identified with a real-linear functional on the real vector space <math>i \mathfrak{h}</math>.

For example, let <math>\mathfrak{g} = \mathfrak{su}(n)</math> and take <math>\mathfrak h \subset \mathfrak g</math> the subspace of all diagonal matrices. Note <math>\mathfrak{g}^{\mathbb{C}} = \mathfrak{sl}_n \mathbb{C}</math>. Let <math>e_i</math> be the linear functional on <math>\mathfrak{h}^{\mathbb{C}}</math> given by <math>e_i(H) = h_i</math> for <math>H = \operatorname{diag}(h_1, \dots, h_n)</math>. Then for each <math>H \in \mathfrak{h}^{\mathbb{C}}</math>,
:<math>[H, E_{ij}] = (e_i(H) - e_j(H)) E_{ij}</math>
where <math>E_{ij}</math> is the matrix that has 1 on the <math>(i, j)</math>-th spot and zero elsewhere. Hence, each root <math>\alpha</math> is of the form <math>\alpha = e_i - e_j, i \ne j</math> and the root space decomposition is the decomposition of matrices:<ref>{{harvnb|Knapp|2002|loc=Ch. IV, § 1, Example 1.}}</ref>
:<math>\mathfrak{g}^{\mathbb{C}} = \mathfrak{h}^{\mathbb{C}} \oplus \bigoplus_{i \ne j} \mathbb{C} E_{ij}.</math>