= Solution
Here the <Cartan subalgebra> of a finite-dimensional complex <semisimple Lie algebra> is a maximal toral subalgebra: it is abelian, its adjoint operators are simultaneously diagonalizable, and its centralizer is itself. These standard properties give the <root-space decomposition>
$$
\boxed{\mathfrak g=\mathfrak h\oplus\bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha,\qquad \mathfrak g_\alpha=\{X\in\mathfrak g:[H,X]=\alpha(H)X\ \forall H\in\mathfrak h\}.}
$$
The <roots of a root system> $\alpha$ are the nonzero <linear functionals> in $\mathfrak h^*$ for which $\mathfrak g_\alpha\ne0$; $\Phi$ is their finite set. Each <root space> is one-dimensional. The zero <weight space> is the centralizer of $\mathfrak h$, hence exactly $\mathfrak h$. Thus the displayed sum is a direct sum of simultaneous eigenspaces of the <Adjoint representation>. It is the Cartan root-space decomposition meant here.
The bracket respects weight addition: $[\mathfrak g_\alpha,\mathfrak g_\beta]\subseteq\mathfrak g_{\alpha+\beta}$, where $\mathfrak g_0=\mathfrak h$ and a missing weight means the zero space. This follows by applying the <Jacobi identity> with $H\in\mathfrak h$ to the bracket of two root vectors. It explains how the vector-space decomposition encodes the <Lie algebra> structure.
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