A Lie algebra for Fr\"olicher groups

Fr\"olicher spaces form a cartesian closed category which contains the category of smooth manifolds as a full subcategory. Therefore, mapping groups such as C^\infty(M,G) or \Diff(M), but also projective limits of Lie groups are in a natural way objects of that category, and group operations are morphisms in the category. We call groups with this property Fr\"olicher groups. One can define tangent spaces to Fr\"olicher spaces, and in the present article we prove that, under a certain additional assumption, the tangent space at the identity of a Fr\"olicher group can be equipped with a Lie bracket. We discuss an example which satisfies the additional assumption.