= Euclidean subspace of a Cartan subalgebra
{c}
{title2=$\mathfrak h_{\mathbb R}=\operatorname{span}_{\mathbb R}\{h_{\alpha_i}\}$}
The real span of the simple <coroots> inside a complex <Cartan subalgebra> carries a positive-definite restriction of the <Killing form>. Equivalently it consists of the $h$ for which all roots have real values. The formula $\kappa(h,h)=\sum_{\alpha\in\Phi}\alpha(h)^2$ is positive for $h\ne0$, since the roots span the dual. The induced dual <inner product> makes the <root system> Euclidean and makes <root reflections> orthogonal.
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