I'm confused about orthogonal bases with respect to a symmetric bilinear form $\phi$.
Consider the quadratic form
$Q: \mathbb{R^{4}} \rightarrow \mathbb{R}$ defined as:
$Q(\vec{x})=-x_2^2+2x_1x_3+x_4^2$
Which is associated to a symmetric bilinear form $\phi$ defined as:
$\phi(\vec{x},\vec{y})=-x_2y_2+x_4y_4+x_1y_3+x_3y_1$
Then consider this subspace $W= \Big\{ {(x_1,x_2,x_3,x_4) \in \mathbb{R^{4}} \mid x_4=2x_1-x_2+4x_3=0 \Big\} }= \mathscr{L}((1,2,0,0),(0,4,1,0))$
Determine a basis $\mathscr{C}=(\vec{c_1},\vec{c_2})$ of $W$ orthogonal with respect to $\phi$.
(The answer is $\mathscr{C}=((1,2,0,0),(-\frac{7}{4},\frac{1}{2},1,0))$)
I started considering $\vec{c_1}=(1,2,0,0)$ but then I don't know how to determine a second vector of $W$ which is orthogonal to it.
My idea was to apply Gram-Schmidt process using $\phi$ instead of the standard dot product but it doesn't work and I can't understand why!
In general what is the method to determine an orthogonal basis of a subspace with respect to $\phi$?
Thanks a lot for your help