I am trying to understand simple groups arising from finite orthogonal groups; these are complicated among groups of Lie type in the sense that one should be careful about characteristic of field (even/odd), dimension of space (even/odd) and type of bilinear form. It also requires (as most books write) that one have to pass to quadratic forms from bilinear form according to characteristic of field. I don't know whether we can stick only to non-degenerate symmetric bilinear form.
I tried to understand this in the following way in which I do not specify characteristic of field, dimension of space, structure of bilinear form etc., and I want to know whether it is correct.
Let $V$ be a finite dimensional vector space over a finite field $F$.
Let $(x,y)\mapsto B(x,y)\in F$ be a bilinear form on $V$ such that it is symmetric and non-degenerate (i.e. $B(x,y)=B(y,x)$ and $B(x,y)=0$ for all $y$ implies $x=0$.)
Let $$G=\{T:V\rightarrow V: B(Tx,Ty)=B(x,y) \forall x,y\in V\}.$$
Then $G$ is a subgroup of $GL(V)$. Consider subgroup $[G,G]$ and then the quotient $$[G,G] / (center).$$ This quotient group is simple with some exceptions on dimension and characteristic of the field. Excluding exceptions, this covers all simple groups arising from orthogonal groups; am I right?