= \langle w,w \rangle = 0$. The second equation implies $\langle x,w \rangle = \langle y,v \rangle = 0$. Then the subspace spanned by $x$ and $w$ is totally isotropic with respect to $\langle \cdot, \cdot \rangle$, and this implies it has dimension $1$, so $x$ and $w$ are linearly dependent. Similarly, also $y$ and $v$ are linearly dependent.

We can partition the boundary in five subsets. The nomenclature of the various types of points will become clear in Section \[bond\_geometric\].

### Vertex. {#compactification:boundary:vertex}

For any real point in $B$, we have $v = w = x = y = 0$. The only nonzero coordinate is $r$, so we have a unique real point $v_0 = (0: \ldots : 0 : 1)$ in $B$, called the *vertex*. A computer algebra computation shows that this is a point of multiplicity $20$ on $X$, but we do not need this fact.

### Inversion Points. {#compactification:boundary:inversion}

Consider the matrix $N := r M + 2y x^t$ (it will be clear later in the discussion why we choose this expression for $N$). A boundary point $\beta$ with $M \ne 0$ and $N \ne 0$ is called an *inversion point*. In this case we have 