Integral Motivic Cohomology of $BSO_{4}$

Motivic cohomology is powerful tool in algebraic geometry with associated realization maps giving important information about the relations between cohomological invariants of schemes and their classifying spaces. The problem of computing general cohomological invariants of these classifying spaces is ongoing. Most relevant to this paper is (1) Totaro's construction of the Chow ring of a classifying space in general and his use of this to study symmetric groups in arXiv:math/9802097, (2) Guillot's similar examination for the Lie groups $G_{2}$ and $Spin(7)$ in arXiv:math/0508122, (3) Field's computation of the Chow ring of $BSO(2n,\mathbb{C})$ in arXiv:math/0411424, and (4) Yagita's work on the $\mathbb{Z}_{2}$-motivic cohomology of $BSO_{4}$ and $BG_{2}$ in [Yag10]. The work presented in this paper covers the computation of the motivic cohomology of $BSO_{4}$ with integral coefficients. The primary approach draws on methods laid out by Guillot and Yagita (arXiv:math/0508122, [Yag10]). These results lay the groundwork for future work, most immediately the analogous computation for $BG_{2}$ ([Por21]).