Let $W(B_n)$ be a Weyl group of type $B_n$ and $SBT (n)$ the set of standard bitableaux of size $n$. Similar to Robinson-Schensted correspondences, I know that there exists a map $W(B_n) \to SBT (n) \times SBT (n)$ such that the image of $W(B_n)$ is the set of the same shape pairs of standard bitableaux.

But, I don't know what the map $W(B_n) \to SBT (n) \times SBT (n)$ is. In particular, $W(B_2) \to SBT (2) \times SBT (2)$? Here \begin{align*} W(B_2) \cong \left\{ \binom{1,2,-1,-2}{1,2,-1,-2}, \binom{1,2,-1,-2}{2,1,-2,-1}, \binom{1,2,-1,-2}{-1,2,1,-2}, \binom{1,2,-1,-2}{1,-2,-1,2}, \binom{1,2,-1,-2}{-2,1,2,-1}, \binom{1,2,-1,-2}{2,-1,-2,1}, \binom{1,2,-1,-2}{-2,-1,2,-1}, \binom{1,2,-1,-2}{-1,-2,1,2}\right\}. \end{align*}

Let $W(B_n)$ be a Weyl group of type $B_n$ and $SBT (n)$ the set of standard bitableaux of size $n$. Similar to Robinson-Schensted correspondences, I know that there exists a map $W(B_n) \to SBT (n) \times SBT (n)$ such that the image of $W(B_n)$ is the set of the same shape pairs of standard bitableaux.

But, I don't know what the map $W(B_n) \to SBT (n) \times SBT (n)$ is. In particular, $W(B_2) \to SBT (2) \times SBT (2)$? Here \begin{align*} W(B_2) \cong & \left\{ \begin{pmatrix} 1 & 2 & -1 & -2 \\ 1 & 2 & -1 & -2 \end{pmatrix}, \begin{pmatrix} 1 & 2 & -1 & -2 \\ 2 & 1 & -2 & -1 \end{pmatrix}, \begin{pmatrix} 1 & 2 & -1 & -2 \\ -1 & 2 & 1 & -2 \end{pmatrix}, \right. \\ & \left. \begin{pmatrix} 1 & 2 & -1 & -2 \\ 1 & -2 & -1 & 2 \end{pmatrix}, \begin{pmatrix} 1 & 2 & -1 & -2 \\ -2 & 1 & 2 & -1 \end{pmatrix}, \begin{pmatrix} 1 & 2 & -1 & -2 \\ 2 & -1 & -2 & 1 \end{pmatrix}, \right. \\ & \left. \begin{pmatrix} 1 & 2 & -1 & -2 \\ -1 & -2 & 1 & 2 \end{pmatrix}, \begin{pmatrix} 1 & 2 & -1 & -2 \\ -2 & -1 & 2 & 1 \end{pmatrix}\right\}. \end{align*}