  As natural extensions of the boson realizations of the su(2)- and the
su(1,1)-algebra, the so(4)- and the so(3,1)-algebras are presented in the form
of boson realizations with four kinds of boson operators. For each algebra, two
forms are discussed. One is constructed in terms of two sets of the boson
operators which play a role of spherical tensor with rank 1/2. The other is
based on the ranks 1 and 0. As a possible application, the Runge-Lenz-Pauli
vector, which is famous in the hydrogen atom, is derived with some aspects.
