  We consider invariant covariant derivatives on reductive homogeneous spaces
corresponding to the well-known invariant affine connections. These invariant
covariant derivatives are expressed in terms of horizontally lifted vector
fields on the Lie group. This point of view allows for a characterization of
parallel vector fields along curves. Moreover, metric invariant covariant
derivatives on a reductive homogeneous space equipped with an invariant
pseudo-Riemannian metric are characterized. As a by-product, a new proof for
the existence of invariant covariant derivatives on reductive homogeneous
spaces and their the one-to-one correspondence to certain bilinear maps is
obtained.
