Bilinear expansions of KP multipair correlators in BKP correlators

In earlier work, Schur lattices of KP and BKP $\tau$-functions, denoted $\pi_{\lambda}(g) ({\bf t})$ and $\kappa_{\alpha} (h)({\bf t}_B)$, respectively, defined as fermionic vacuum expectation values, were associated to every GL$(\infty)$ group element $\hat{g}$ and SO$(\tilde{\mathcal{H}}^\pm, Q_\pm)$ group element $\hat{h}$. The elements of these lattices are labelled by partitions $\lambda$ and strict partitions $\alpha$, respectively. It was shown how the former may be expressed as finite bilinear sums over products of the latter. In this work, we show that two-sided KP tau functions corresponding to any given $\hat{g}$ may similarly be expressed as bilinear combinations of the corresponding two-sided BKP tau functions.