We have the quadratic form on $Q(x,y,z) = x^2 - y^2$ on $=\mathbb{R}^3$ considered as an $\mathbb{R}$-vector space. We want to calculate the set of vectors $v$ such that there exists a diagonal basis for $Q$ that contains $v$.
I do not know how to think about this problem. I have worked out that the quadratic form has matrix $\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ in the standard basis $\{e_1, e_2, e_3\}$. From here we can see that the maximal dimension of a subspace on which $Q$ is positive-definite is $1$. And the maximal dimension of a subspace on which $Q$ is negative definite is $1$. And the maximal dimension of a subspace on which $Q$ is $0$ is 2.
We can think about some vectors that are definitely going to be in the set. To start off, $\lambda e_1, \lambda e_2, \lambda e_3$ (the x, y and z axes) will all be in the set.
I don't know how to rule vectors out from being candidates to be in the set. For example, if I took the vector $(1,1,0)$, how can I decide whether it is possible for there to be a diagonal basis of $Q$ that contains $(1,1,0)$?
How can we solve this? And how can we prove the solution is correct and that no vectors have been missed?