Let's say I'm given the Cartan matrix of D4 : $$\begin{matrix} 2 & -1 & 0 & 0 & & \\ -1 & 2 & -1 & -1 \\ 0 & -1 & 2 & 0 & \\ 0 & -1 & 0 & 2 \\ \end{matrix}$$ Let's name the elements of each copy of $sl(2)$ inside our algebra $(X_i, Y_i, H_i), i=1 \dots 4$, with $X_i$ the positive Chevalley generators and $H_i$ the Cartan subalgebra. We know from the Serre relations that for example $[X_1, X_2] \neq 0$ while $[X_1, X_3] = 0$. However, how can we know that $[X_1, [X_2, X_3]]$ is or isn't equal to $0$ ? Without computing the roots, I mean, in which case we can deduce it. But can we know it from the Cartan matrix itself ? And what about $[X_3, [X_1, [X_2, X_4]]]$ or the more general case ? Thanks a lot for your answers.
EDIT : Maybe a more well-defined question would be to ask how to know the dimension of the Lie subalgebra generated by the $X_i$ from the Serre relations.