Let $H$ be a simple algebraic group of type $\mathbf{G}_2$ over $\mathbb{C}$.
Let $\rho$ be the standard 7-dimensional complex representation
$$ \rho\colon H=\mathbf{G}_2\to \mathrm{SO}_7.$$
We consider the corresponding homomorphism
$$H\to \mathrm{Spin}_7=G.$$
We regard $H$ as a subgroup of $G$.
Let $T_H\subset H$ and $T_G\subset G$ be compatible maximal tori.
We obtain a homomorphism of cocharacter groups
$$\rho_*\colon X_*(T_H)\to X_*(T_G).$$
We choose compatible Borel subgroups $B_H$ and $B_G$ such that $T_H\subset B_H\subset H$
and  $T_G\subset B_G\subset G$, then we obtain  bases consisting of simple coroots in $X_*(T_H)$ and $X_*(T_G)$.
Since we have bases, we can associate with $\rho_*$ an integral  $3\times 2$ matrix.

> **Question.** How can one compute this matrix?