  We construct left invariant special K\"ahler structures on the cotangent
bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted
cartesian product of two special K\"ahler Lie algebras according to two linear
representations by infinitesimal K\"ahler transformations. We also exhibit a
double extension process of a special K\"ahler Lie algebra which allows us to
get all simply connected special K\"ahler Lie groups with bi-invariant
symplectic connections. All Lie groups constructed by performing this double
extension process can be identified with a subgroup of symplectic (or K\"ahler)
affine transformations of its Lie algebra containing a nontrivial $1$-parameter
subgroup formed by central translations. We show a characterization of left
invariant flat special K\"ahler structures using \'etale K\"ahler affine
representations, exhibit some immediate consequences of the constructions
mentioned above, and give several non-trivial examples.
