<s>[INST] Which Quadric Surface Is $-9x'^2+3y'^2+6z'^2=1$ 
Classify the quadric surface
$$3x^2-2y^2-z^2-4xy-12yz-8xz=1$$

I have found the eigenvalues which are $-9,6,3$
So it is $$-9x'^2+3y'^2+6z'^2=1$$
According to a page I have $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ is an elliptic hyperboloid of one sheet and $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1$ is an elliptic hyperboloid of two sheets.
So which is $3x^2-2y^2-z^2-4xy-12yz-8xz=1$?
 [/INST] Note that $-9x'^2+3y'^2+6z'^2=1$ is equivalent to
$$\frac{X^2}{(1/\sqrt{6})^2}+\frac{Y^2}{(1/\sqrt{3})^2}-\frac{Z^2}{(1/3)^2}=1$$
where $X=z'$, $Y=y'$ and $Z=x'$ (we swapped $x'$-axis and $z'$-axis).
So it is a one-sheet elliptic hyperboloid. 
</s>