= Solution
The weights of the seven-dimensional irreducible $G_2$ representation are zero and the six short roots. In the $A_2$ <weight lattice>, the three positive short roots are
$$
\alpha=\omega_2-\omega_1,\qquad
\alpha+\beta=\omega_1,\qquad
2\alpha+\beta=\omega_2.
$$
Together with their negatives, they split into the weight sets of the two dual three-dimensional <Fundamental representations of sl3>; the zero weight supplies a trivial representation. By the <Restriction of the seven-dimensional G2 representation to long-root A2>,
$$
\boxed{V_7\!\downarrow_{\mathfrak{sl}_3}
\cong V(\omega_1)\oplus V(\omega_2)\oplus V(0).}
$$
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