  During the last two decades, great efforts have been devoted to the
calculation of the local theta correspondence for reductive dual pairs.
However, uniform formulas remain elusive for real dual pairs of type I. The
purpose of this paper is twofold: to formulate an explicit version of induction
principle for dual pairs $(O(p,q),Sp(2n,\mathbb{R}))$ with $p + q$ even, and to
apply it to obtain a complete and explicit description of the local theta
correspondence when $p + q = 4$. Our approach is very elementary by analysis on
the infinitesimal characters and K-types under the correspondence.
