The infinitesimal cone of a totally positive semigroup

Given a complex reductive linear algebraic group split over $\mathbb {R}$ with a fixed pinning, it is shown that all elements of the Lie algebra $\mathfrak {g}$ infinitesimal to the totally positive subsemigroup $G_{\ge 0}$ of $G$ lie in the totally positive cone $\mathfrak {g}_{\ge 0}\subset \mathfrak {g}$.