  The Mordell-Weil lattices (MW lattices) associated to rational elliptic
surfaces are classified into 74 types. Among them, there are cases in which the
MW lattice is none of the weight lattices of simple Lie algebras or direct sums
thereof. We study how such "non-Cartan MW lattices" are realized in the
six-dimensional heterotic/F-theory compactifications. In this paper, we focus
on non-Cartan MW lattices that are torsion free and whose associated
singularity lattices are sublattices of $A_7$. For the heterotic string
compactification, a non-Cartan MW lattice yields an instanton gauge group $H$
with one or more $U(1)$ group(s). We give a method for computing massless
spectra via the index theorem and show that the $U(1)$ instanton number is
limited to be a multiple of some particular non-one integer. On the F-theory
side, we examine whether we can construct the corresponding threefold
geometries, i.e., rational elliptic surface fibrations over $P^1$. Except for
some cases, we obtain such geometries for specific distributions of instantons.
All the spectrum derived from those geometries completely match with the
heterotic results.
