  To give a criterion for the integrability of Banach-Lie triple systems, we
follow the construction of the period group of a Lie algebra and define the
period group of a Lie triple system as an analogous concept. We show that a Lie
triple system is integrable if and only if its period group is discrete. Along
the way, we see how to turn the path and the loop space of a pointed symmetric
space into pointed symmetric spaces.
