What rotations are used as a reduction step in Kenig-Ruiz-Sogge's uniform Sobolev estimate?

Question

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?

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[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.

Answer 1

It turns out that (1) Yes, I will need to use 'nonstandard' rotations, but (2) the hyperbolic ones suffice. This is because (as already in question body) the usual Euclidean rotations in $\mathbb R^j$ and $\mathbb R^{n-j}$ turns the vector $a\in \mathbb R^n$ essentially into a 2D vector which we can consider as living in a hyperbolic plane. (in the event $j=0,n$ there is nothing to do.) So my problem stems from not understanding any hyperbolic geometry at all :) and it could well be that a passer-by could have easily pointed this to me had I better phrased my problem. Alas!

In any case: the question body reduces the issue into the study of $b=(|a'|,0,\dots,0,|a''|)$ where $c = \sqrt{Q(a)}$ with $a=(a',a'')\in \mathbb R^j \times \mathbb R^{n-j}$ and $ Q(b) = |a'|^2 - |a''|^2$.

After the reduction that lets us assume $a\not\in\operatorname{Char} P $, we either have $Q(a)>0$ or $Q(a)<0$, or equivalently $|a'|^2>|a''|^2$ or $|a'|^2<|a''|^2$. In the first case (the second entirely analogous) we can set $$ c = \sqrt{Q(b)},\quad \theta = \cosh^{-1}|a'/c|$$ so that $$ \tilde b = b/c = (\sinh \theta) e_1 + (\cosh \theta) e_n. $$ Then the "hyperbolic rotation" $$ R(x_1,\dots,x_n) = (x_1\cosh\theta - x_n \sinh \theta, x_2,\dots, x_{n-1}, -x_1\sinh \theta + x_n \cosh \theta)$$ can be easily seen by a direct calculation to preserve $Q$ and map $\tilde b$ to $e_n$.

(The relationship to the differential operator $a\cdot \nabla$ comes by using the rotated function $u_R(x)=u(R^Tx)$, for which $a\cdot\nabla u_R(x) = (\partial_n u)(R^Tx).$)