I think I have understood the bulk of the paper [KRS], but one of the parts I cannot understand is when the authors reduce Theorem 2.1 (p.332) into Proposition 2.1 (p.335). I can understand all the reductions except for the one using a rotation.

In this paper, $Q$ is a nonsingular quadratic form on $\mathbb R^n$, $n\ge 3$, given by$$Q(\xi) = -\xi_{1}^2-\dots-\xi_{\mathstrut j}^2 + \xi_{j+1}^2 + \dots \xi_{\mathstrut n}^2.$$ They apparently say: when $a\neq 0$ is such that $Q(a)\neq 0$, it can be rotated while preserving the form of $Q$ so that $a\in \operatorname{span}((0,\dots,0,1))$ or $a\in \operatorname{span}((1,0,\dots,0))$. Why is this true?

With a rotation in the first $j$ components and then another in the remaining ones, I can make $a\in \operatorname{span}((1,0,\dots,0),(0,\dots,0,1))$. So 'clearly' I guess I'm missing some sort of rotation that intermingles these two parts of $Q$. But what rotations of this type preserve $Q$?

Later in the paper near the end, the author specifies in the case where $Q(D)$ is the wave operator $\partial_t^2 -\Delta$ that one needs hyperbolic rotations. Do I need to figure out an appropriate group of "rotations" for $Q$ in the more general case?

[KRS] *Kenig, C. E.; Ruiz, A.; Sogge, C. D.*, **Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators**, Duke Math. J. 55, 329-347 (1987). ZBL0644.35012.