Restricted root systems are always reduced

Given: The Axiom of Choice; the real semisimple Lie algebra $\mathfrak{s}\mathfrak{u}(2,1)$ with $J=\mathrm{diag}(1,1,-1)$, the Cartan involution $\theta (X)=-X^{\ast }$, the maximal abelian subspace $\mathfrak{a}=\mathbb{R}H\subseteq {\mathfrak{p}}_0$ for $H=E_{13}+E_{31}$, and its restricted-root system $\mathrm{\Sigma }\subseteq {\mathfrak{a}}^{\ast }$ with the functionals $f$ given by $f(H)=1$.

The Axiom of Choice is The Axiom of Choice; it is the hypothesis required by the restricted-root suppliers used in [L3].

Restricted roots and restricted-root spaces are the nonzero functionals $\lambda \in {\mathfrak{a}}^{\ast }$ with ${\mathfrak{g}}_0^{\lambda }=\{X:[H,X]=\lambda (H)X$ for all $H\in \mathfrak{a}\}\mathrm{\ne }0$, and the multiplicity $m_{\lambda }$ is ${\dim }_{\mathbb{R}}{\mathfrak{g}}_0^{\lambda }$ (Restricted root and restricted root space).

A reduced crystallographic root system satisfies, in addition to the reflection and integrality conditions, the reducedness condition $\mathbb{R}\alpha \cap \mathrm{\Phi }=\{\alpha ,-\alpha \}$ for every $\alpha \in \mathrm{\Phi }$ (Reduced crystallographic Euclidean root system).

For ${\mathfrak{g}}_0=\mathfrak{s}\mathfrak{u}(2,1)=\{X\in M_3(\mathbb{C}):X^{\ast }J+JX=0,\mathrm{tr}X=0\}$ with $J=\mathrm{diag}(1,1,-1)$, $\theta (X)=-X^{\ast }$ and $\mathfrak{a}=\mathbb{R}H$, $H=E_{13}+E_{31}$, the restricted-root system is $\mathrm{\Sigma }=\{\pm f,\pm 2f\}$ with $f(H)=1$ and multiplicities $m_{\pm f}=2$, $m_{\pm 2f}=1$; moreover $\mathfrak{s}\mathfrak{u}(2,1)$ is a real form of ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$, hence a real semisimple Lie algebra (Restricted root systems may be nonreduced, Restricted root space decomposition).