Let π°(π―(1)) denote the enveloping algebra of the two-dimensional nonabelian Lie algebra π―(1) over a base field π. We study the maximal abelian ad-nilpotent (mad) associative subalgebras and finite-dimensional Lie subalgebras of π°(π―(1)). We first prove that the set of noncentral elements of π°(π―(1)) admits the Dixmier partition, π°(π―(1))βπ=β 5i=1Ξi, and establish characterization theorems for elements in Ξi, i=1,3,4. Then we determine the elements in Ξi, i=1,3 , and describe the eigenvalues for the inner derivation ad Bx,xβΞi, i=3,4 . We also derive other useful results for elements in Ξi, i=2,3,4,5 . As an application, we find all framed mad subalgebras of π°(π―(1)) and determine all finite-dimensional nonabelian Lie algebras that can be realized as Lie subalgebras of π°(π―(1)) . We also study the realizations of the Lie algebra π―(1) in π°(π―(1)) in detail.