3 columns in align environment

Question

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*}

Answer 1

The easiest way to align everything is probably to put all of the lines in the same align environment.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
[P_\mu,P_\nu] &= 0   & \xrightarrow{\hspace*{4cm}}  && 
    [\tilde{H},\tilde{H}] &= 0 \\
    &&&& [\tilde{P_i},\tilde{H}] &= 0 \\
    &&&& [\tilde{P_i},\tilde{P_j}] &= 0 \\ 
[M_{\mu \nu },P_\rho] &= 2 \eta_{\rho[\nu}P_{\mu]}   & \xrightarrow{\hspace*{4cm}}  &&  
    [\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 \\ 
[M_{\mu \nu},M_{\rho\sigma}] &= 4\eta_{[\mu[\rho}M_{\sigma]\nu ]}  & \xrightarrow{\hspace*{4cm}}  && 
    [\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{align*}
\end{document}

Answer 2

With alignat* you control the spacing between columns. I added some vertical spacing between the three groups:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{alignat*}{3}
[P_\mu,P_\nu] &= 0 &&{} \xrightarrow{\hspace*{4cm}}{} & [\tilde{H},\tilde{H}] &= 0 \\
  & & & & [\tilde{P_i},\tilde{H}] &= 0 \\
  & & & & [\tilde{P_i},\tilde{P_j}]& = 0 \\[1.5ex]
[M_{\mu \nu },P_\rho] &= 2 \eta_{\rho[\nu}P_{\mu]} & & \xrightarrow{\hspace*{4cm}}{} &
 [\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 \\[1.5ex]
[M_{\mu \nu},M_{\rho\sigma}] &= 4\eta_{[\mu[\rho}M_{\sigma]\nu ]} & & \xrightarrow{\hspace*{4cm}} & [\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{alignat*}

\end{document}