We study the family of Cartan-Schouten connections on Lie groups, parameterized by $\lambda\in[0,1]$, whose geodesics through the identity are one-parameter subgroups. We compute their curvature, torsion, parallel transport, and geodesics, and develop Euler-Poincaré and Lie-Poisson reduction for mechanical systems via these connections, unifying the “minus” and “plus” cases. These inspire us to introduce a connection-dependent variational principle where the Lagrangian is expressed in terms of the parallel-transported velocity, leading to an integro-differential Euler-Lagrange equation that explicitly involves torsion and curvature memory terms. The general framework is illustrated on two concrete examples: the Heisenberg group, where the equations simplify to an ODE system, and the rotation group SO(3), where the integro-differential system is solved numerically via a Magnus expansion.