Compact roots form a reduced crystallographic root system

Fix the invariant inner product of [L1]. Differentiation gives $⟨[X,U],V⟩=-⟨U,[X,V]⟩$. Thus for every ideal $I$, its orthogonal complement is again an ideal: if $U\perp I$, then $⟨[X,U],V⟩=-⟨U,[X,V]⟩=0$ for $V\in I$. Repeatedly splitting proper invariant subspaces in finite dimension proves complete reducibility of the adjoint module. Now, with its premise verified, [L2] gives $\mathfrak{g}=\mathfrak{z}\oplus \mathfrak{s}$ with $\mathfrak{s}$ semisimple. Also $⟨Z,[U,V]⟩=-⟨[U,Z],V⟩=0$ for $Z\in \mathfrak{z}$, so this direct sum is orthogonal.

The toral algebra $\mathfrak{t}$ is maximal abelian. Indeed, for any abelian $\mathfrak{a}\supseteq \mathfrak{t}$, [L3] makes $A={\exp }_G(\mathfrak{a})$ a connected abelian subgroup. Its closure is a connected compact subgroup; continuity of division and commutators extends the subgroup and abelian identities to the closure. The closed-subgroup theorem makes it an embedded torus. It contains $T$ because naturality and local invertibility of ${\exp }_T$ show that $A$ contains an identity neighborhood in $T$, whose generated subgroup is open and closed in connected $T$. Its tangent algebra contains $\mathfrak{a}$, by differentiating the curves ${\exp }_G(tV)$ inside that embedded subgroup. Maximality of $T$ therefore gives $\mathfrak{a}=\mathfrak{t}$. In particular $\mathfrak{z}\subseteq \mathfrak{t}$.

Let ${\mathfrak{t}}^{\mathrm{\prime }}=\mathfrak{t}\cap \mathfrak{s}$. Steps 1.1 and 1.2 give the orthogonal splitting $\mathfrak{t}=\mathfrak{z}\oplus {\mathfrak{t}}^{\mathrm{\prime }}$; any element of $\mathfrak{s}$ centralizing ${\mathfrak{t}}^{\mathrm{\prime }}$ centralizes $\mathfrak{t}$, so lies in ${\mathfrak{t}}^{\mathrm{\prime }}$. The Killing form $B_{\mathfrak{s}}$ is negative definite: in a real orthonormal basis the skew-adjoint matrix $A={\mathrm{ad}}_U{\mathrm{\mid }}_{\mathfrak{s}}$ satisfies $\mathrm{tr}{A}^2=-{\sum }_{a,b}{A}_{ab}^2$, which is negative for $U\mathrm{\ne }0$ because $\mathfrak{s}$ is centerless. In the same real basis the complexified Killing form is its complex-bilinear extension, hence nondegenerate. By [L2], ${\mathfrak{s}}_{\mathbb{C}}$ is complex semisimple.

The group center is closed, since it is the intersection of the closed sets on which conjugation by each fixed element is the identity. Thus $Z(G{)}^0$ is a compact connected abelian Lie subgroup by [L3]. The product $Z(G{)}^{0}T$ is a subgroup because the first factor is central; it is compact and connected as a continuous image of the compact connected product, and is abelian. It is closed in Hausdorff $G$, hence an embedded torus containing $T$. Maximality forces $Z(G{)}^0\subseteq T$. Its conjugation action is trivial, so each root character takes value $1$ on it. Infinitesimally every root vanishes on $\mathfrak{z}$ by step 1.2 and the bracket formula of [L6].

Put $\mathfrak{h}={\mathfrak{t}}_{\mathbb{C}}^{\mathrm{\prime }}$. The commuting skew-adjoint operators ${\mathrm{ad}}_H$, $H\in {\mathfrak{t}}^{\mathrm{\prime }}$, become normal operators for the Hermitian extension of the real inner product, hence simultaneously diagonalize by [L4]. Their common zero eigenspace in ${\mathfrak{s}}_{\mathbb{C}}$ is $\mathfrak{h}$, since real and imaginary parts of a commuting vector lie in ${\mathfrak{t}}^{\mathrm{\prime }}$ by step 2.1. The eigenvalue functions extend complex-linearly to $\mathfrak{h}$. If $U$ normalizes $\mathfrak{h}$, decompose it into the simultaneous eigenspaces. In $[H,U]$, each nonzero-weight component is its component of $U$ times its nonzero functional evaluated at $H$. The condition $[H,U]\in \mathfrak{h}$ for every $H$ forces each such component to vanish. Thus the normalizer is $\mathfrak{h}$. It is abelian, hence nilpotent, so is a Cartan subalgebra by [L4]. The hypotheses of [L5] have now all been established.

Apply [L5] to $({\mathfrak{s}}_{\mathbb{C}},\mathfrak{h})$. Adding the central summand gives a decomposition of ${\mathfrak{g}}_{\mathbb{C}}$ with zero space ${\mathfrak{t}}_{\mathbb{C}}$ and the nonzero root spaces of ${\mathfrak{s}}_{\mathbb{C}}$. The $T$-operators preserve these spaces: $T$ fixes $\mathfrak{t}$ and its adjoint action commutes with their infinitesimal operators. Equivalently, use the character decomposition [L6]; differentiating it and comparing with this infinitesimal decomposition shows that its nontrivial characters correspond bijectively to the Lie-algebra roots, extended by zero on ${\mathfrak{z}}_{\mathbb{C}}$. The correspondence is injective because differentials determine characters, and surjective because a nonzero infinitesimal root space contains a nonzero character eigenspace.

On $i{\mathfrak{t}}^{\mathrm{\prime }}$, the extended Killing form is positive definite by step 2.1. The roots are real-valued there by [L6]. For each root, real linear algebra therefore gives a unique $H_{\alpha }\in i{\mathfrak{t}}^{\mathrm{\prime }}$ with $B(H_{\alpha },H)=\alpha (H)$ on $i{\mathfrak{t}}^{\mathrm{\prime }}$; complex linearity extends the equation to $\mathfrak{h}$, so this is the Killing-dual vector in [L5]. Its nonzero real norm shows that $h_{\alpha }$ also lies in $i{\mathfrak{t}}^{\mathrm{\prime }}$. By [L5] the real coroot span is a real form of $\mathfrak{h}$, of the same dimension as $i{\mathfrak{t}}^{\mathrm{\prime }}$, and thus equals $i{\mathfrak{t}}^{\mathrm{\prime }}$. Consequently [L5] supplies precisely the reduced crystallographic Euclidean root system on $(i{\mathfrak{t}}^{\mathrm{\prime }}{)}^{\ast }$ for the dual of this positive Killing metric. Reflection invariance, integrality and spanning follow with no change of scale or character convention. Extend its functionals by zero on $i\mathfrak{z}$; since the decomposition of $i\mathfrak{t}$ is orthogonal, their span is the dual subspace annihilating the central directions, identified with their orthogonal complement.

Steps 2.2 and 5.1 prove the claims. Here vanishing on the central torus means being the trivial character, or value zero in the additive $\mathbb{R}\mathrm{/}\mathbb{Z}$ convention. If $\mathfrak{s}=0$, then $\mathfrak{g}=\mathfrak{z}=\mathfrak{t}$ and there are no nonzero weights; the empty root set is the rank-zero root system in the zero vector space. All zero-dimensional cases are included. AC is used as stated in [A1], not inferred from compactness alone.