In the textbook "Kac-Moody Groups, their Flag Varieties and Representation Theory" by Shrawan Kumar, he defines the minimal Kac-Moody groups, $\mathcal{G}_{min}$, as the group generated by a Torus, $T$, and one parameter groups, $\mathcal{U}_{\alpha}$, corresponding to real root spaces of the Lie algebra. Later, he defines the action of this group on its Lie algebra, denoted by Ad.
My question: I am trying to understand how the Torus acts on the Cartan subalgebra. I think it is acting by identity but I am not sure. In other words, is it true that $\text{Ad}t\left(h\right)=h\,\,\text{for } t\in T,\,h\in\mathfrak{h}$? Thank you in advance!