Factor of similitude group for a bilinear form

Definition
Let $$k$$ be a field, $$V$$ a (usually finite-dimensional) vector space over $$k$$, and $$b$$ a bilinear form on $$V$$. The factor of similitude group for $$b$$ is the subgroup of the multiplicative group of $$k$$ comprising those $$\lambda$$ such that there exists an invertible linear transformation $$A$$ satisfying:

$$b(Av,Aw) = \lambda b(v,w) \ \forall \ v,w \in V$$.

We typically assume $$b$$ to be nondegenerate, though this is not necessary to make sense of the definition.

Facts

 * All squares are in the factor of similitude group, because if $$\alpha$$ is in the multiplicative group of $$k$$, then the factor of similitude for scalar multiplication by $$\alpha$$ is $$\alpha^2$$
 * For $$b$$ nondegenerate, the factor of similitude group is contained in the subgroup comprising $$n^{th}$$ roots of squares, because the determinant (which is the $$n^{th}$$ power of the factor of similitude) must be a square.