Let $G=D_8=\langle a,b\mid a^4=b^2=1,b^{-1}ab=a^{-1}\rangle$. The character table of $D_8$ is known and is
Let
$$U:=\bigg\{\sum\limits_{1\leq i<j\leq 4} a_{ij}x_ix_j\mid a_{ij}\in\mathbb{C}\bigg\},$$
be the vector space spanned up by quadratic monomials with different indices. Then I define the representation $\rho:G\to GL(U)$ by
$$ \rho(a)P(x_1,x_2,x_3,x_4)=P(x_2,x_3,x_4,x_1), \quad \rho(b)P(x_1,x_2,x_3,x_4)=P(x_4,x_3,x_2,x_1), $$
for all $P\in U$.
Then $\rho$ defines a representation of $D_8$ which is the direct sum of two times $V_0$, $V_2,V_3$ and $V$. I have found bases of $V_0,V_2,V_3$ but I want to find a basis of $V$. Are there systematic ways to find this basis apart from just trying?