[proofplan] We prove the inclusion by testing the bracket of two weight vectors against an arbitrary element of the Cartan subalgebra. The Jacobi identity rewrites the action of $h \in \mathfrak{h}$ on $[x,y]$ in terms of the known actions on $x \in \mathfrak{g}_\alpha$ and $y \in \mathfrak{g}_\beta$. This shows that $[x,y]$ is a weight vector of weight $\alpha+\beta$, and the stated convention handles the case where $\alpha+\beta$ is not a root or zero. [/proofplan] [step:Test the bracket against an arbitrary Cartan element] Fix $\alpha,\beta \in \mathfrak{h}^*$, and let $x \in \mathfrak{g}_\alpha$ and $y \in \mathfrak{g}_\beta$. By the definition of the root spaces, for every $h \in \mathfrak{h}$ we have \begin{align*} [h,x] &= \alpha(h)x, \\ [h,y] &= \beta(h)y. \end{align*} We must prove that $[x,y] \in \mathfrak{g}_{\alpha+\beta}$, meaning that for every $h \in \mathfrak{h}$, \begin{align*} [h,[x,y]] = (\alpha+\beta)(h)[x,y]. \end{align*} [/step] [step:Use the Jacobi identity to compute the weight of the bracket] Let $h \in \mathfrak{h}$. Applying the Jacobi identity in the form \begin{align*} [h,[x,y]] = [[h,x],y] + [x,[h,y]], \end{align*} and substituting the root-space relations for $x$ and $y$, we obtain \begin{align*} [h,[x,y]] &= [[h,x],y] + [x,[h,y]] \\ &= [\alpha(h)x,y] + [x,\beta(h)y] \\ &= \alpha(h)[x,y] + \beta(h)[x,y] \\ &= (\alpha(h)+\beta(h))[x,y] \\ &= (\alpha+\beta)(h)[x,y]. \end{align*} Here we used bilinearity of the Lie bracket over $\mathbb{C}$ in the third equality. [/step] [step:Conclude that the bracket lies in the root space of the summed weight] Since the identity \begin{align*} [h,[x,y]] = (\alpha+\beta)(h)[x,y] \end{align*} holds for every $h \in \mathfrak{h}$, the element $[x,y]$ belongs to the $(\alpha+\beta)$-weight space. By the definition and convention for $\mathfrak{g}_{\alpha+\beta}$, this says exactly that \begin{align*} [x,y] \in \mathfrak{g}_{\alpha+\beta}. \end{align*} Because $x \in \mathfrak{g}_\alpha$ and $y \in \mathfrak{g}_\beta$ were arbitrary, we conclude \begin{align*} [\mathfrak{g}_\alpha,\mathfrak{g}_\beta] \subseteq \mathfrak{g}_{\alpha+\beta}. \end{align*} [/step]