two column AND multi-line align

Question

I am trying to get the 3rd equality (from the top, the equations for delta \phi^i) to align.

\begin{align*}
U(x)  &= e^{i\theta^a(x)T^a}             &  U(x) &= e^{-\theta^a(x)(t_a)^i{}_j}  \\ 
 [T^a,T^b] &= if^{abc}T^c  &  [t^a,t^b] &= f_{ab}{}^c t^c \\
\delta \phi_i &= i\theta^a(T^a)_{ij} \phi_j  &  \begin{split}
 \delta \phi^i &= -\theta^a(t_a)^i{}_j \phi^j \\
&= -\theta^af_{aj}{}^i \phi^j \\
&= \theta^af_{ja}{}^i \phi^j
\end{split} \\
A_\mu  &= A_\mu{}^a T^a            &  A_\mu &=  A_\mu{}^a t_a \\ 
A_\mu  &\rightarrow U(x) A_\mu U(x)^{-1} -
\frac{i}{g} [\partial_\mu U(x)] U(x)^{-1} &  A_\mu  &\rightarrow U(x) A_\mu U(x)^{-1} -
\frac{1}{g} [\partial_\mu U(x)] U(x)^{-1}
\end{align*}

Here are two related questions

How to align a set of multiline equations

Aligning two multiline equations

Answer 1

Use aligned with [t] option?

Is this what you're after?

\documentclass{extarticle}
\usepackage{amsmath}
\begin{document}
\begin{align*}
    U(x)  &= e^{i\theta^a(x)T^a}             &  U(x) &= e^{-\theta^a(x)(t_a)^i{}_j}  \\ 
    [T^a,T^b] &= if^{abc}T^c  &  [t^a,t^b] &= f_{ab}{}^c t^c \\
    \delta \phi_i &= i\theta^a(T^a)_{ij} \phi_j  &  
    \delta \phi^i & 
    \begin{aligned}[t]
         &= -\theta^a(t_a)^i{}_j \phi^j \\
        &= -\theta^af_{aj}{}^i \phi^j \\
        &= \theta^af_{ja}{}^i \phi^j
    \end{aligned} \\
    A_\mu  &= A_\mu{}^a T^a            &  A_\mu &=  A_\mu{}^a t_a \\ 
    A_\mu  &\rightarrow U(x) A_\mu U(x)^{-1} -
    \frac{i}{g} [\partial_\mu U(x)] U(x)^{-1} &  A_\mu  &\rightarrow U(x) A_\mu U(x)^{-1} -
    \frac{1}{g} [\partial_\mu U(x)] U(x)^{-1}
\end{align*}
\end{document}

Answer 2

• you can write your equation with use only align* environment
• equation is wider than text width, consequently it spill out from right text border:

(red lines indicate page layout)

\documentclass{extarticle}
\usepackage{amsmath}
%---------------- show page layout. don't use in a real document!
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%

\begin{document}
    \begin{align*}
U(x)        
    & = e^{i\theta^a(x)T^a} &  U(x)         & = e^{-\theta^a(x)(t_a)^i{}_j}  \\
[T^a,T^b]   
    & = if^{abc}T^c         &  [t^a,t^b]    & = f_{ab}{}^c t^c  \\
\delta\phi_i    
    & = i\theta^a(T^a)_{ij}\phi_j   &
            \delta \phi^i   & = -\theta^a(t_a)^i{}_j \phi^j     \\ 
    &&                      & = -\theta^af_{aj}{}^i \phi^j      \\
    &&                      & = \theta^af_{ja}{}^i \phi^j       \\
A_\mu  
    & = A_\mu{}^a T^a       &  
            A_\mu           &=  A_\mu{}^a t_a               \\
A_\mu  
    & \rightarrow U(x) A_\mu U(x)^{-1} - \frac{i}{g} [\partial_\mu U(x)] U(x)^{-1} &  
                    A_\mu   & \rightarrow U(x) A_\mu U(x)^{-1} 
                                        - \frac{1}{g} [\partial_\mu U(x)] U(x)^{-1}
    \end{align*}
\end{document}

• this problem you can solve on two ways, for example:
  • enlarge text area for example with use of the `geometry package
  • break last equation in each column

\documentclass{extarticle}
\usepackage{amsmath}
%---------------- show page layout. don't use in a real document!
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%

\begin{document}
    \begin{align*}
U(x)        
    & = e^{i\theta^a(x)T^a} &  U(x)         & = e^{-\theta^a(x)(t_a)^i{}_j}  \\
[T^a,T^b]   
    & = if^{abc}T^c         &  [t^a,t^b]    & = f_{ab}{}^c t^c  \\
\delta\phi_i    
    & = i\theta^a(T^a)_{ij}\phi_j   &
            \delta \phi^i   & = -\theta^a(t_a)^i{}_j \phi^j 
                              = -\theta^af_{aj}{}^i \phi^j      \\
    &&                      & = \theta^af_{ja}{}^i \phi^j       \\
A_\mu  
    & = A_\mu{}^a T^a       &  
            A_\mu           &=  A_\mu{}^a t_a                   \\
A_\mu  
    & \rightarrow U(x) A_\mu U(x)^{-1}  &   
                    A_\mu   & \rightarrow U(x) A_\mu U(x)^{-1}  \\
    &\qquad - \frac{i}{g} [\partial_\mu U(x)] U(x)^{-1} & 
                            &\qquad - \frac{1}{g} [\partial_\mu U(x)] U(x)^{-1}
    \end{align*}
\end{document}