I have a question about minuscule cocharacters, which might sound trivial to the experts:

Let $G$ be a smooth affine group scheme over $\mathbb{Z}_{p}$. Furthermore, consider a cocharacter $$\mu\colon \mathbb{G}_{m,W(k)}\rightarrow G_{W(k)},$$ where $k$ is some finite field. Consider the action of $\mathbb{G}_{m,W(k)}$ on $G_{W(k)},$ given by $x\ast g=\mu(x)^{-1}g\mu(x).$ We have the unipotent group-scheme associated to $\mu,$ denoted by $U^{+}(\mu).$ Furthermore we can consider the adjoint representation $Ad(\mu^{-1})$ of $\mathbb{G}_{m,W(k)}$ on the Lie-algebra, which induces a $\mathbb{Z}$-grading on it: $$\mathfrak{g}=\bigoplus_{n\in \mathbb{Z}} \mathfrak{g}_{n}.$$

The question: Assume that $G$ is reductive now. Is the requirement that $\mathfrak{g}_{n}=0$ for $n\geq 2$ equivalent to $\mu$ being minuscule, i.e. $\mathfrak{g}_{n}=0$ for all $|n|\geq 2?$

Thanks!