  Bounded symmetric domains carry several natural invariant metrics, for
example the Carath\'eodory, Kobayashi or the Bergman metric. We define another
natural metric, from generalized Hilbert metric defined in [FGW20], by
considering the Borel embedding of the domain as an open subset of its dual
compact Hermitian symmetric space and then its Harish-Chandra realization in
projective spaces. We describe this construction on the four classical families
of bounded symmetric domains and compute both this metric and its associated
Finsler metric. We compare it to Carath\'eodory and Bergman metrics and show
that, except for the complex hyperbolic space, those metrics differ.
