Restricted root and restricted root space

Let ${\mathfrak{g}}_0$ be a finite-dimensional real semisimple Lie algebra, let $\mathfrak{a}$ be a maximal split abelian subspace of the Cartan decomposition ${\mathfrak{g}}_0={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$ (Maximal split abelian subspace and real rank), and let ${\mathfrak{a}}^{\ast }={\mathrm{Hom}}_{\mathbb{R}}(\mathfrak{a},\mathbb{R})$ be its dual space. For $\lambda \in {\mathfrak{a}}^{\ast }$ put ${\mathfrak{g}}_0^{\lambda }=\{X\in {\mathfrak{g}}_0:[H,X]=\lambda (H)X\text{ for every }H\in \mathfrak{a}\}\mathrm{.}$ Each ${\mathfrak{g}}_0^{\lambda }$ is a real linear subspace of ${\mathfrak{g}}_0$, because the bracket is bilinear and the condition is linear in $X$, and ${\mathfrak{g}}_0^{\lambda }$ depends only on the functional $\lambda$; the assignment $\lambda \mapsto {\mathfrak{g}}_0^{\lambda }$ is compatible with the bracket, $[{\mathfrak{g}}_0^{\lambda },{\mathfrak{g}}_0^{\mu }]\subseteq {\mathfrak{g}}_0^{\lambda +\mu }(\lambda ,\mu \in {\mathfrak{a}}^{\ast }),$ by the Jacobi identity, as proved in Restricted root space decomposition.

A restricted root of ${\mathfrak{g}}_0$ with respect to $\mathfrak{a}$ is a nonzero functional $\lambda \in {\mathfrak{a}}^{\ast }$ with ${\mathfrak{g}}_0^{\lambda }\mathrm{\ne }0$; the set of restricted roots is denoted $\mathrm{\Sigma }=\mathrm{\Sigma }({\mathfrak{g}}_0,\mathfrak{a})=\{\lambda \in {\mathfrak{a}}^{\ast }:\lambda \mathrm{\ne }0,{\mathfrak{g}}_0^{\lambda }\mathrm{\ne }0\}\mathrm{.}$ For $\lambda \in \mathrm{\Sigma }$ the subspace ${\mathfrak{g}}_0^{\lambda }$ is the restricted-root space of $\lambda$, and its real dimension $m_{\lambda }={\dim }_{\mathbb{R}}{\mathfrak{g}}_0^{\lambda }$ is the multiplicity of $\lambda$; it is a positive integer.

The case $\lambda =0$ is not a restricted root but is part of the notation: ${\mathfrak{g}}_0^0=\{X\in {\mathfrak{g}}_0:[H,X]=0\text{ for every }H\in \mathfrak{a}\}=Z_{{\mathfrak{g}}_0}(\mathfrak{a})$ is the centralizer of $\mathfrak{a}$ in ${\mathfrak{g}}_0$, and $\mathfrak{a}\subseteq {\mathfrak{g}}_0^0$ because $\mathfrak{a}$ is abelian. For $H\in \mathfrak{a}$ the space ${\mathfrak{g}}_0^{\lambda }$ is contained in the eigenspace of the endomorphism $\mathrm{ad}H$ for the eigenvalue $\lambda (H)$.

The restricted roots of ${\mathfrak{g}}_0$ relative to $\mathfrak{a}$ are the real-algebra analogue of the roots of a complex semisimple Lie algebra relative to a Cartan subalgebra: the role of the Cartan subalgebra is played by the maximal abelian subspace $\mathfrak{a}$ of ${\mathfrak{p}}_0$, which diagonalizes the commuting family $\{\mathrm{ad}H:H\in \mathfrak{a}\}$ of self-adjoint endomorphisms of ${\mathfrak{g}}_0$. The resulting restricted-root space decomposition, the finiteness of $\mathrm{\Sigma }$, the identity ${\mathfrak{g}}_0^0=\mathfrak{a}\oplus Z_{{\mathfrak{k}}_0}(\mathfrak{a})$, and the sign and bracket properties of the spaces ${\mathfrak{g}}_0^{\lambda }$ are proved in Restricted root space decomposition.