# Help with definition of hyperbolic angle

I am currently reading about the Hyperbolic Space in the book Metric Spaces of Non-Positive Curvature of Martin R. Bridson and André Haefliger and they define the space $$\mathbb{E}^{n,1}$$ as $$\mathbb{R}^{n+1}$$ equipped with the following bilinear form $$\left[u,v\right] = \left(\sum_{i=1}^{n}u_{i}v_{i}\right) - u_{n+1}v_{n+1}$$ and the Hyperbolic n-Space $$\mathbb{H}^{n}$$ as: $$\mathbb{H}^{n} = \{x \in \mathbb{E}^{n,1} : \left[x,x\right] = -1, x_{n+1}>0\}.$$ Now given $$A \in \mathbb{H}^{n}$$ and a vector $$u\in \mathbb{E}^{n,1}$$ such that $$\left[u,u\right] = 1$$ and $$\left[u,A\right] = 0$$, then a hyperbolic segment is an image of an compact interval of the path $$c : \mathbb{R} \to \mathbb{H}^{n}$$, $$c\left(t\right) = \left(\cosh{t}\right)A+\left(\sinh{t}\right)u$$, we call $$u$$ the initial vector of the segment. They define the angle between two hyperbolic segments issuing from a point of $$\mathbb{H}^{n}$$, with initial vectors $$u$$ and $$v$$ is the unique number $$\alpha \in \left[0,\pi\right]$$ such that $$\cos{\alpha} = \left[u,v\right].$$

Ok, here is my problem, I don't see how the angle is well-defined, for this, I would need to have $$\left|\left[u,v\right]\right|\leq 1$$, I tried to use the definition of $$\left[\cdot,\cdot\right]$$ and using Cauchy-Schwarz inequality some times but it was useless, any help?

I hope you agree that if $$u, v$$ live in a definite subspace then $$[u, v] = \cos\alpha$$ for some $$0 \leq \alpha \leq \pi$$; they are unit vectors and this is just the usual expression for angle in a Euclidean space. Notice that $$[\cdot,\cdot]$$ is a form with signature $$(n,1)$$, meaning that any orthogonal basis for $$\mathbb E^{n,1}$$ must have $$n$$ vectors $$[x,x] > 0$$ and 1 vector with $$[x,x] < 0$$. In particular, for any $$y$$ with $$[y,y] < 0$$ we have the orthogonal decomposition $$\mathbb E^{n,1} = \mathbb Ry\oplus y^\perp$$ with $$y^\perp$$ the orthogonal complement of $$y$$, and this $$y^\perp$$ must be a positive-definite space.
By definition $$u, v \in A^\perp$$ with $$[A,A] = -1 < 0$$, so $$u, v$$ live in a definite space and the $$\alpha$$ in $$\cos\alpha = [u,v]$$ is well-defined.