I would like the equal signs in the 3rd column of the following to be aligned.
I can't see why it is not working based on two column AND multi-line align
\begin{align*}
[P_\mu,P_\nu] &= 0 & \xrightarrow{\hspace*{4cm}} &&
\begin{aligned}[t]
[\tilde{H},\tilde{H}] &= 0 \\
[\tilde{P_i},\tilde{H}] &= 0 \\
[\tilde{P_i},\tilde{P_j}] &= 0
\end{aligned} \\
[M_{\mu \nu },P_\rho] &= 2 \eta_{\rho[\nu}P_{\mu]} & \xrightarrow{\hspace*{4cm}} &&
\begin{aligned}[t]
[\tilde{K_i},\tilde{H}] &= \tilde{P_i} \\
[\tilde{K_i},\tilde{P_j}] &= \delta_{ij} \tilde{H} \\
[\tilde{J_{ij}},\tilde{P_k}] &= 2 \delta_{k[j}\tilde{P_{i]}} \\
[\tilde{J_{ij}},\tilde{H}] &= 0
\end{aligned} \\
[M_{\mu \nu},M_{\rho\sigma}] &= 4\eta_{[\mu[\rho}M_{\sigma]\nu ]} & \xrightarrow{\hspace*{4cm}} &&
\begin{aligned}[t]
[\tilde{K_i},\tilde{K_j}] &= 0 \\
[\tilde{J_{ij}},\tilde{K_k}] &= 0 \\
[\tilde{J_{ij}},\tilde{J_{kl}}] &= 4\delta_{[i[k}\tilde{J}_{l]j ]}
\end{aligned} \\
\end{align*}