Assume characteristic 0. I do not know how much of this extends to finite characteristic.

For the Lie algebras, you have $\mathbf u=C_{\mathbf u}(\mathbf t)\oplus\sum \mathbf u_\alpha$ for $\alpha\in\Delta(\mathbf g,\mathbf t)$ (roots of $\mathbf g$ w.r.t. $\mathbf t$). 

If $\alpha(T)\neq\beta(T)$ and $\alpha(T)\neq 0$ for all roots $\alpha,\beta\in\Delta(\mathbf g,\mathbf t_{max})$ and some element $T\in\mathbf t$ (such an element is called regular), the roots w.r.t. $\mathbf t$ are in bijection with those w.r.t. $\mathbf t_{max}$, the root spaces are 1-dimensional and $\mathbf u$ is a sum of root spaces.