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?

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?

Note that I need this matrix only modulo 2.