Additive actions on toric varieties

By an additive action on an algebraic variety $X$ of dimension $n$ we mean a regular action $\mathbb{G}_a^n \times X \to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. We prove that if a complete toric variety $X$ admits an additive action, then it admits an additive action normalized by the acting torus. Normalized additive actions on a toric variety $X$ are in bijection with complete collections of Demazure roots of the fan of $X$. Moreover, any two normalized additive actions on $X$ are isomorphic.