Isotropic vectors of quadratic form How can i find all isotropic vectors of quadratic form
$x_{1}^2+4x_{2}^2+8x_{3}^2-4x_{1}x_{2}+8x_{1}x_{3}-14x_{2}x_{3}$

I don't understand how to approach.
Use Lagrange's Reduction (complete the square). I'll use $\;x,y,z\;$ for ease:

$$x^2+4y^2+8z^2-4xy+8xz-14yz=(x^2-2x(2y+4z))+4y^2+8z^2-14yz=$$

$$=\left(x-2y-4z\right)^2-\color{red}{4y^2}-16yz-16z^2+\color{red}{4y^2}+8z^2-14yz=$$

$$=\left(x-2y-4z\right)^2-8\left(z^2+\frac{15}2yz\right)=(x-2y-4z)^2-8\left(z+\frac{15}4y\right)^2+\frac{225}2y^2$$

Thus, your quadratic is equivalent to $\;\tilde x^2+\tilde y^2-\tilde z^2\;$ (signature $\,(2,1)\,$) , so it actually is a cone.

$$$$
Integer and primitive:
  $$ \pm \; \; \left(4 u^2 + 2uv-4v^2, \; 4 u^2 - 2 v^2, \; u^2 \right)  $$
 $$ \pm \; \;  \left(8 u^2 - 2uv-2v^2, \; 8 u^2 -  v^2, \; 2 u^2 \right)  $$

====================================================================

$$ P^T H P = D  $$
$$\left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
 - 4 &amp; 0 &amp; 1 \\ 
 \frac{ 3 }{ 2 }  &amp; 1 &amp;  \frac{ 1 }{ 8 }  \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrr} 
1 &amp;  - 2 &amp; 4 \\ 
 - 2 &amp; 4 &amp;  - 7 \\ 
4 &amp;  - 7 &amp; 8 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrr} 
1 &amp;  - 4 &amp;  \frac{ 3 }{ 2 }  \\ 
0 &amp; 0 &amp; 1 \\ 
0 &amp; 1 &amp;  \frac{ 1 }{ 8 }  \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
0 &amp;  - 8 &amp; 0 \\ 
0 &amp; 0 &amp;  \frac{ 1 }{ 8 }  \\ 
\end{array}
\right) 
$$
tells us that we can take any column vector $C=(u,v,w)^T$ with $8u^2 + w^2 = 64 v^2,$ multiply that vector to get $X=PC,$ then $X^THX = 0$   

=======================================

$$ H = \left( 
\begin{array}{rrr} 
1 &amp;  - 2 &amp; 4 \\ 
 - 2 &amp; 4 &amp;  - 7 \\ 
4 &amp;  - 7 &amp; 8 \\ 
\end{array}
\right) 
$$
$$  D_0 = H  $$
$$ E_j^T D_{j-1} E_j = D_j  $$
$$  P_{j-1} E_j = P_j  $$
$$ E_j^{-1} Q_{j-1}  = Q_j  $$
$$  P_j Q_j = Q_j P_j = I  $$
$$ P_j^T H P_j = D_j  $$
$$ Q_j^T D_j Q_j = H  $$

$$ H = \left( 
\begin{array}{rrr} 
1 &amp;  - 2 &amp; 4 \\ 
 - 2 &amp; 4 &amp;  - 7 \\ 
4 &amp;  - 7 &amp; 8 \\ 
\end{array}
\right) 
$$

==============================================

$$ E_{1} = \left( 
\begin{array}{rrr} 
1 &amp; 2 &amp; 0 \\ 
0 &amp; 1 &amp; 0 \\ 
0 &amp; 0 &amp; 1 \\ 
\end{array}
\right) 
$$
$$  P_{1} = \left( 
\begin{array}{rrr} 
1 &amp; 2 &amp; 0 \\ 
0 &amp; 1 &amp; 0 \\ 
0 &amp; 0 &amp; 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{1} = \left( 
\begin{array}{rrr} 
1 &amp;  - 2 &amp; 0 \\ 
0 &amp; 1 &amp; 0 \\ 
0 &amp; 0 &amp; 1 \\ 
\end{array}
\right) 
, \; \; \; D_{1} = \left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 4 \\ 
0 &amp; 0 &amp; 1 \\ 
4 &amp; 1 &amp; 8 \\ 
\end{array}
\right) 
$$

==============================================

$$ E_{2} = \left( 
\begin{array}{rrr} 
1 &amp; 0 &amp;  - 4 \\ 
0 &amp; 1 &amp; 0 \\ 
0 &amp; 0 &amp; 1 \\ 
\end{array}
\right) 
$$
$$  P_{2} = \left( 
\begin{array}{rrr} 
1 &amp; 2 &amp;  - 4 \\ 
0 &amp; 1 &amp; 0 \\ 
0 &amp; 0 &amp; 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{2} = \left( 
\begin{array}{rrr} 
1 &amp;  - 2 &amp; 4 \\ 
0 &amp; 1 &amp; 0 \\ 
0 &amp; 0 &amp; 1 \\ 
\end{array}
\right) 
, \; \; \; D_{2} = \left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
0 &amp; 0 &amp; 1 \\ 
0 &amp; 1 &amp;  - 8 \\ 
\end{array}
\right) 
$$

==============================================

$$ E_{3} = \left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
0 &amp; 0 &amp; 1 \\ 
0 &amp; 1 &amp; 0 \\ 
\end{array}
\right) 
$$
$$  P_{3} = \left( 
\begin{array}{rrr} 
1 &amp;  - 4 &amp; 2 \\ 
0 &amp; 0 &amp; 1 \\ 
0 &amp; 1 &amp; 0 \\ 
\end{array}
\right) 
, \; \; \; Q_{3} = \left( 
\begin{array}{rrr} 
1 &amp;  - 2 &amp; 4 \\ 
0 &amp; 0 &amp; 1 \\ 
0 &amp; 1 &amp; 0 \\ 
\end{array}
\right) 
, \; \; \; D_{3} = \left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
0 &amp;  - 8 &amp; 1 \\ 
0 &amp; 1 &amp; 0 \\ 
\end{array}
\right) 
$$

==============================================

$$ E_{4} = \left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
0 &amp; 1 &amp;  \frac{ 1 }{ 8 }  \\ 
0 &amp; 0 &amp; 1 \\ 
\end{array}
\right) 
$$
$$  P_{4} = \left( 
\begin{array}{rrr} 
1 &amp;  - 4 &amp;  \frac{ 3 }{ 2 }  \\ 
0 &amp; 0 &amp; 1 \\ 
0 &amp; 1 &amp;  \frac{ 1 }{ 8 }  \\ 
\end{array}
\right) 
, \; \; \; Q_{4} = \left( 
\begin{array}{rrr} 
1 &amp;  - 2 &amp; 4 \\ 
0 &amp;  -  \frac{ 1 }{ 8 }  &amp; 1 \\ 
0 &amp; 1 &amp; 0 \\ 
\end{array}
\right) 
, \; \; \; D_{4} = \left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
0 &amp;  - 8 &amp; 0 \\ 
0 &amp; 0 &amp;  \frac{ 1 }{ 8 }  \\ 
\end{array}
\right) 
$$

==============================================

$$ P^T H P = D  $$
$$\left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
 - 4 &amp; 0 &amp; 1 \\ 
 \frac{ 3 }{ 2 }  &amp; 1 &amp;  \frac{ 1 }{ 8 }  \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrr} 
1 &amp;  - 2 &amp; 4 \\ 
 - 2 &amp; 4 &amp;  - 7 \\ 
4 &amp;  - 7 &amp; 8 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrr} 
1 &amp;  - 4 &amp;  \frac{ 3 }{ 2 }  \\ 
0 &amp; 0 &amp; 1 \\ 
0 &amp; 1 &amp;  \frac{ 1 }{ 8 }  \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
0 &amp;  - 8 &amp; 0 \\ 
0 &amp; 0 &amp;  \frac{ 1 }{ 8 }  \\ 
\end{array}
\right) 
$$
$$ Q^T D Q = H  $$
$$\left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
 - 2 &amp;  -  \frac{ 1 }{ 8 }  &amp; 1 \\ 
4 &amp; 1 &amp; 0 \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrr} 
1 &amp; 0 &amp; 0 \\ 
0 &amp;  - 8 &amp; 0 \\ 
0 &amp; 0 &amp;  \frac{ 1 }{ 8 }  \\ 
\end{array}
\right) 
\left( 
\begin{array}{rrr} 
1 &amp;  - 2 &amp; 4 \\ 
0 &amp;  -  \frac{ 1 }{ 8 }  &amp; 1 \\ 
0 &amp; 1 &amp; 0 \\ 
\end{array}
\right) 
 = \left( 
\begin{array}{rrr} 
1 &amp;  - 2 &amp; 4 \\ 
 - 2 &amp; 4 &amp;  - 7 \\ 
4 &amp;  - 7 &amp; 8 \\ 
\end{array}
\right) 
$$