I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on $\mathfrak{g}=\mathrm{Lie}(G)$ by the adjoint action. Then $\mathfrak{g}$ decomposes into a direct sum of eigenspaces with eigenvalues given by roots in the character group $X(T)$. These roots form an irreducible abstract root system inside of $\mathbb{R}\otimes_{\mathbb{Z}}X(T)$. Choosing a base of the root system $\Phi$ leads to a unique highest root, and the coxeter number of $G$ can be defined as
$$h(G)=1+\sum a_i$$
where the $a_i$ are the coefficients of the simple roots in the linear combination expressing the highest root.
Now, if I have a semisimple or reductive group, then the root system of $G$ is not necessarily irreducible, so I'm not sure there is a unique highest root with respect to some chosen base. However, in the literature I still see reference to the coxeter number of a reductive group. How is this defined? Thanks!