Corollary 2

Suppose that $\mathbf{A}$ is a field and that $\mathbf{E}$ is finite-dimensional. In order that a projector $\nu$ in $\mathbf{E}$ be such that $\nu(\mathbf{E})$ and $\nu(0)$ are orthogonal (and consequently non-isotropic), it is necessary and sufficient that $\nu = \nu^*$.

It suffices to apply prop. 4 to the involution $u = 1 - 2\nu$.

A projector satisfying the condition of Corollary 2 is called an orthogonal projector for $\Phi$.