Let $u_1 = (a_1, b_1, c_1, d_1)$ and $u_2 = (a_2, b_2, c_2, d_2)$ be any vectors in $\Bbb R^4$ Which inner product axioms do not hold with the definition
$\left \langle u_1, u_2\right\rangle = a_1a_2 + 2b_1b_2 - c_1c_2 + 2d_1d_2$
Now I believe that symmetry and positivity would hold. It seems that homogeneity could hold, as the scalar doesn't affect the order of multiplication. This would mean additivity is the only one that does not hold. If that is correct, I am having trouble constructing it to show it fails. Cause wouldn't $(a_1 + a_2)w_1 + 2(b_1 + b_2)w_2 - (c_1 + c_2)w_3 + 2(d_1 + d_2)w_4$ still be an instance of $\left \langle u_1, w\right\rangle + \left \langle u_2, w\right\rangle $ ?
It is not an inner product because the inner product of $(0,0,1,0)$ with itself is $-1 <0$. $ \langle u, u \rangle \geq 0$ is a necessary condition for $ \langle ., . \rangle$ to be an inner product.
[Linearity in each variable is true in this case].