The root set is a reduced crystallographic root system

Orthogonality of distinct root spaces and nondegeneracy of the Killing form imply that $B$ restricts to a nondegenerate pairing ${\mathfrak{g}}_{\alpha }\times {\mathfrak{g}}_{-\alpha }\to \mathbb{C}$. Choose $e_{\alpha },f_{\alpha }$ with $B(e_{\alpha },f_{\alpha })\mathrm{\ne }0$. Then Opposite root spaces bracket to the Killing-dual line gives $[e_{\alpha },f_{\alpha }]=B(e_{\alpha },f_{\alpha })H_{\alpha }$. The scalar $\alpha (H_{\alpha })$ is nonzero: otherwise the generated three-dimensional algebra would be solvable, so its adjoint action could be triangularized, making $\mathrm{ad}[e_{\alpha },f_{\alpha }]$ nilpotent; but this element lies in $\mathfrak{h}$ and acts semisimply, forcing it to be zero, a contradiction. After rescaling, $e_{\alpha },f_{\alpha }$ and $h_{\alpha }:=2H_{\alpha }\mathrm{/}\alpha (H_{\alpha })$ form an ${\mathfrak{s}\mathfrak{l}}_2$-triple.

The subspace $\mathbb{C}{H}_{\alpha }\oplus {⨁}_{k\mathrm{\ne }0}{\mathfrak{g}}_{k\alpha }$ is a finite-dimensional module for this ${\mathfrak{s}\mathfrak{l}}_2$. Its zero-weight space is the line $\mathbb{C}{H}_{\alpha }$, so complete reducibility of ${\mathfrak{s}\mathfrak{l}}_2$-modules leaves one irreducible summand with even weights. Consequently its weight-$2$ space ${\mathfrak{g}}_{\alpha }$ is one-dimensional. Since $[e_{\alpha },e_{\alpha }]=0$, the raising operator kills that weight-$2$ space, so it is the highest weight: no $k\alpha$ with $\mathrm{\mid }k\mathrm{\mid }\ge 2$ is a root.

For roots $\alpha ,\beta$, the root-string space $V_{\alpha ,\beta }:={⨁}_{k\in \mathbb{Z}}{\mathfrak{g}}_{\beta +k\alpha }$ is a finite-dimensional ${\mathfrak{s}\mathfrak{l}}_2$-module. Since its nonzero weight spaces are one-dimensional by step 2.1, the vector in ${\mathfrak{g}}_{\beta }$ lies in one irreducible summand. The $h_{\alpha }$-weight $\beta (h_{\alpha })=2(\beta ,\alpha )\mathrm{/}(\alpha ,\alpha )$ is therefore an integer, and symmetry of the weights in that summand supplies the nonzero opposite-weight space ${\mathfrak{g}}_{\beta -\beta (h_{\alpha })\alpha }$. Thus $s_{\alpha }(\beta )=\beta -\beta (h_{\alpha })\alpha$ is again a root.

The roots span ${\mathfrak{h}}^{\ast }$, since an element of $\mathfrak{h}$ annihilated by every root would commute with all of $\mathfrak{g}$ and hence be zero. On the real span of the coroots, the Killing form is positive definite: $B(h,h)=\mathrm{tr}({\mathrm{ad}}_h^2)$ is the sum of squares of the real root eigenvalues and is positive for $h\mathrm{\ne }0$. Finiteness comes from the finite root decomposition; step 2.1 gives reducedness; and step 3.1 gives crystallographic integrality and reflection stability. Hence $\mathrm{\Phi }$ is a finite reduced crystallographic root system, and all its root spaces are one-dimensional.