  We prove that a selfdual $GL_3$-Galois representation constructed by van
Geemen and Top is isomorphic to a quadratic twist of the symmetric square of
the Tate module of an elliptic curve. This is an application of our refinement
of the Faltings-Serre method to $3$-dimensional Galois representations with the
ground field not equal to $\mathbb{Q}$. The proof makes use of the
Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.
