In the case of a two-leg Hubbard ladder we present a procedure which allows the exact deduction of the ground state for the four-particle problem in an arbitrary large lattice system, in a tractable manner, which involves only a reduced Hilbert space region containing the ground state. In the presented case, the method leads to nine analytic, linear and coupled equations providing the ground state. The procedure which is also applicable to few particle problems and other systems is based on an r-space representation of the wavefunctions and construction of symmetry adapted orthogonal basis wave vectors describing the Hilbert space region containing the ground state. Once the ground state is deduced, a complete quantum-mechanical characterization of the studied state can be given. Since the analytic structure of the ground state becomes visible during the use of the method, it is important not only to the understanding of theoretical aspects connected to exact descriptions or potential numerical approximation scheme developments, but is also relevant for a large number of potential technological application possibilities placed between nano-devices and quantum calculations, where the few particle behaviour and deep understanding are important key aspects.