  We prove the following result on the timelike spherical Hilbert geometry of
simplices: Let $\Delta_2$ be a simplex on the 2-sphere and $\tilde{\Delta}_2$
the antipodal simplex. We show that the timelike spherical Hilbert geometry
associated with the pair $\Delta_2, \tilde{\Delta}_2$ is isometric to a union
of six copies of vector spaces equipped with a timelike norm, isometrically and
transitively acted upon by the group $\mathbb{R}_{>0}^2 \times
\mathbb{Z}_3\times \mathbb{Z}_2$. This is a timelike spherical analogue of a
well-known result (due to Busemann) stating that the Hilbert metric of a
Euclidean simplex is isometric to a metric induced by a normed vector space. At
the same time, this gives a new example of timelike space.
