  Associated to a symmetric space there is a canonical connection with zero
torsion and parallel curvature. This connection acts as a binary operator on
the vector space of smooth sections of the tangent bundle, and it is linear
with respect to the real numbers. Thus the smooth section of the tangent bundle
together with the connection form an algebra we call the connection algebra.
The constraints of zero torsion and constant curvature makes the connection
algebra into a Lie admissible triple algebra. This is a type of algebra that
generalises pre-Lie algebras, and it can be embedded into a post-Lie algebra in
a canonical way that generalises the canonical embedding of Lie triple systems
into Lie algebras. The free Lie admissible triple algebra can be described by
incorporating triple-brackets into the leaves of rooted (non-planar) trees.
