  In this paper we study the invariant metrizability and projective
metrizability problems for the special case of the geodesic spray associated to
the canonical connection of a Lie group. We prove that such canonical spray is
projectively Finsler metrizable if and only if it is Riemann metrizable. This
result means that this structure is rigid in the sense that considering
left-invariant metrics, the potentially much larger class of projective Finsler
metrizable canonical sprays, corresponding to Lie groups, coincides with the
class of Riemann metrizable canonical sprays. Generalisation of these results
for geodesic orbit spaces are given.
