I was reading section 2 (Clifford Modules) of the Notes on Spinors by Deligne, and am a little puzzled by Proposition 2.2. In this proposition, it is assumed $V$ is a complex vector space and $Q$ a quadratic form. The proof begins immediately by saying that $V$ is isomorphic to $L \oplus L^*$ with $Q(l+\alpha)=\langle{\alpha,l}\rangle$. He later goes on to decompose this direct direct sum as a sum over one dimensional spaces: $\sum_i L_i \oplus L^*_i$ - which I gather is what is meant by saying $Q$ is a hyperbolic form.
The reference is to Bourbaki Algebra Ch. 9, which I see tells us that we require $Q$ to be 'neutral' (which is essentially the statement that $V \cong L \oplus L^*$ with $L$ and $L^*$ isotropic). But in the notes, he makes no such assumption.
What exactly does he mean here? What is this isomorphism? Is he using the fact that $V$ is a complex vector space, and so has a complex structure under which we can decompose $V$ into a direct sum of eigenspaces of $\pm i$? As you can see, I am a little confused - perhaps someone could help clear this up a little.
Thanks!