diagonalization of quadratic form over R

Question

I have the quadratic form

$$q\left(x_1,\:x_2,\:x_3,x_4\right)=2x_1x_4-6x_2x_3$$

$q:\:R^4-R^4$

I need to :

1. express it as a sum of quadrants (quadratic terms)

2. find a base B of $R^4$ in which q is diagonal.

3. find sub-space $U$ of $R^4$ of maximal dimension, that $\forall \ \ 0 \neq x\in U,\:q\left(u\right)>0 $

I see that $$\left[q\right]_E=\begin{pmatrix}0&0&0&1\\ 0&0&-3&0\\ 0&-3&0&0\\ 1&0&0&0\end{pmatrix}$$

Is it diagnolizable or am I wrong in its representation ? Can I express it as sum of quadratic elements at all ? Ho do I approach to that last question ?

I tried the method, which says "do raw and col elementary operations on $[q]_E$ to get a diagonal matrix, and then repeat the columns operations them on the I matrix and i supposed to get P, but some why I couldn't diagonilize $[q]_E$, I mean technically/algebraic. (maybe I missed it somehow dk)"

Answer 1

$$4ab=(a+b)^2-(a-b)^2$$

Hence $$2x_1x_4-6x_2x_3=\tfrac{1}{2}(x_1+x_4)^2-\tfrac{1}{2}(x_1-x_4)^2-\tfrac{3}{2}(x_2+x_3)^2+\tfrac{3}{2}(x_2-x_3)^2$$