  This paper presents new research in infinitesimal algebra by introducing the
concept of an infinitesimal group and exploring its properties and
ramifications. The author investigates first- and second-order subgroups of Lie
groups and demonstrates the use of the second-order infinitesimal group
structure to define a Lie bracket of points intrinsic to the Lie group. This
construction allows for the derivation of a second-order
Baker-Campbell-Hausdorff formula for the infinitesimal group operation and
provides a means to reconstruct the Lie bracket of the Lie algebra of a Lie
group. The author also characterises all second-order infinitesimal group
structures on KL vector spaces as deformations of vector addition by bilinear
maps. The main contribution of the paper is the generalisation of the
previously established correspondence between symmetric affine connections and
second-order infinitesimally affine structures to manifolds with non-symmetric
affine connections via second-order infinitesimal groups.
