Is there a source in which Demazure's function $p$ defined in SGA3, exp. XXI, is calculated?

Suppose that $mathcal{R}=(M,R,M^*,R^*)$ is a root datum. In section 1.2 of SGA3, exp. XXI, Demazure defines the $mathbb Z$-linear map $p:Mto M^*$ by
$$p(x)=sum_{uin R^*}(u,x)u$$
and proves many things about it, especially when $mathcal{R}$ is reduced and irreducible. However, as far as I can see, he never calculates this function. Is it calculated anywhere, say when $mathcal{R}$ is also semi-simple? In particular, I wish to calculate the biggest integer $n$ such that the image of $p$ lies in $nM^*$ when $mathcal R$ is adjoint.

(N.B.: this homomorphism $p$ has no connection with the $p$-morphisms discussed later in the same exposé for prime numbers $p$.)