<p>A loop is called diassociative if any two elements generate a subgroup, an anticommutative algebra is binary Lie if any two elements are contained in a Lie subalgebra. In binary Lie algebras the elements generate one-parameter subgroups in the corresponding Lie groups, so the exponential and logarithm maps are locally well defined. The logarithm of the product of the exponential images defines the multiplication of a local analytic diassociative loop, represented by the classical Baker-Campbell-Hausdorff series. We study a family of binary Lie algebras for which the closed form of the Baker-Campbell-Hausdorff series defines the multiplication function of an analytic diassociative loop on the entire binary Lie algebra. These algebras are semidirect sums of the two-dimensional non-abelian and an abelian Lie algebra, the Lie subalgebras generated by the 2-frames have dimension 2 or 3. We express the group multiplications of the exponential images of elements of the matrix Lie algebras that are isomorphic to these Lie subalgebras. By transforming back to the binary Lie algebra, we express the analytic diassociative loop multiplication. Our result contributes to the Lie theory of diassociative loops, since no fully developed examples of the correspondence between binary Lie algebras and global analytic diassociative loops are known so far.</p>