Remember, a point in \(PC \) is a light ray in \(\mathrm{Im}(\mathbb{O}')\):
a 1-dimensional subspace on which the quadratic form \(Q\) vanishes.

This notion doesn't use the nonassociative algebra structure of split octonions, just the quadratic form \(Q\).

But in fact, we may equivalently define a point in \(PC\) to be a 1-dimensional null subalgebra of \( \mathbb{O}' \): a subspace of \(\mathbb{O}'\) on which the product vanishes.

We may then define a line in \(PC\) to be a collection of 1-dimensional null subalgebras that lies in some 2-dimensional null subalgebra of \(\mathbb{O}'\).