  Hilbert's and Thompson's metric spaces on the interior of cones in
JB-algebras are important examples of symmetric Finsler spaces. In this paper
we characterize the Hilbert's metric isometries on the interiors of cones in
JBW-algebras, and the Thompson's metric isometries on the interiors of cones in
JB-algebras. These characterizations generalize work by Bosch\'e on the Hilbert
and Thompson isometries on symmetric cones, and work by Hatori and Moln\'ar on
the Thompson isometries on the cone of positive self-adjoint elements in a
unital $C^*$-algebra. To obtain the results we develop a variety of new
geometric and Jordan algebraic techniques.
