  A simple systematic procedure to construct the proton-neutron unitary,
$u_{\text{sd}}^{\pi \nu }{(12)}$, orthogonal, $o_{\text{sd}}^{\pi \nu }{(12)}$,
and quasi-spin $\text{su}_{\text{sd}}^{\pi \nu }{(1,1)}$ algebras of the sd
bosonic system is presented. New algebraic substructures of these algebras are
discussed and the explicit formulae for their generators and Casimir operators
are given in the spherical tensor form. The complementarity relationship of the
Casimir operators of the $\text{su}_{\text{sd}}^{\pi \nu }{(1,1)}$ and
$o_{\text{sd}}^{\pi \nu }{(12)}$ is derived. The exact algebraic solutions of
the quantum phase transition Hamiltonian between the $o_{\text{sd}}^{\pi \nu
}{(12)}$ and $u_s^{\pi \nu }{(2)} \otimes u_d^{\pi \nu }{(10)}$ limits has been
considered, for the first time, in the framework of affine
$\text{su}_{\text{sd}}^{\pi \nu }{(1,1)}$ Lie algebra. The low lying energy
spectra of the $\, ^{70}\text{Ge},\, ^{76-78}\text{Se},\,
^{96-98}\text{Mo},\text{and}\, ^{100-102}\text{Ru}$ isotopes are calculated
using the $o_{\text{sd}}^{\pi \nu } {(12)} {\leftrightarrow} u_s^{\pi \nu
}{(2)} \otimes u_d^{\pi \nu }{(10)}$ transition Hamiltonian. The good agreement
of our computation with empirical result in these isotopes emphasizes the
importance of $u_s^{\pi \nu }{(2)} \otimes u_d^{\pi \nu }{(10)}$ limit. With
this addition, symmetry can be extended to many nuclei.
