How to fix the error of align

Question

Actually I am trying too much but I am not able to fix the error after running: Any help is appreciated:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{align*}  
    (f_{xy},f_{yz}) _{[w]}&=[f_{xy},z] - [x, f_{yz}]\\
                                         &=[[x,y]- \sum \alpha_{xy}^v v ,z ] - [x ,[y,z] -\sum\alpha_{yz}^v v ]\\
                                         &=[[x,y,z]]- \sum \alpha_{xy}^v [v,z] - [x,[y,z]] + \sum \alpha_{yz}^v [x,v]\\
\text{Using Jacobi's identity this equals}
                                         &=[[x,z],y] - \sum\alpha_{xy}^v ([v,z] - \sum \alpha_{vz}^u u)\\
                                         &+ \sum \alpha_{yz}^v ( [x,v] - \sum \alpha_{xv}^u u ) - \sum \alpha_{xy}^v \alpha_{vz}^u u + \sum     \alpha_{yz}^v \alpha_{xv}^u u\\
                                        &= [[x,z] - \sum \alpha_{xz}^v v , y ] + \sum \alpha_{xz}^v ([v,y] - \sum \alpha_{vy}^u u)\\
                                        &-\sum \alpha_{xy}^v f_{vz} + \sum \alpha_{yz}^v f_{xv}\\
                                        & -\sum_{u} (\sum \alpha_{xy}^v \alpha_{vz}^u + \sum \alpha_{yz}^v \alpha_{vx}^u + \sum \alpha_{zx}^v \alpha_{vy}^u) u
\end{align*} 
\end{document}

Answer 1

your code works fine, but it seems that you looking for intertext as Heiko Oberdiek suggest in his comment:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
    \begin{align*}
(f_{xy},f_{yz}) _{[w]}&=[f_{xy},z] - [x, f_{yz}]\\
         &=[[x,y]- \sum \alpha_{xy}^v v ,z ] - [x ,[y,z] -\sum\alpha_{yz}^v v ]\\
         &=[[x,y,z]]- \sum \alpha_{xy}^v [v,z] - [x,[y,z]] + \sum \alpha_{yz}^v [x,v]
\intertext{Using Jacobi's identity this equals:}
    &=[[x,z],y] - \sum\alpha_{xy}^v ([v,z] - \sum \alpha_{vz}^u u)\\
    &+ \sum \alpha_{yz}^v ( [x,v] - \sum \alpha_{xv}^u u ) - \sum \alpha_{xy}^v \alpha_{vz}^u u + \sum     \alpha_{yz}^v \alpha_{xv}^u u\\
    &= [[x,z] - \sum \alpha_{xz}^v v , y ] + \sum \alpha_{xz}^v ([v,y] - \sum \alpha_{vy}^u u)\\
    &-\sum \alpha_{xy}^v f_{vz} + \sum \alpha_{yz}^v f_{xv}\\
    & -\sum_{u} (\sum \alpha_{xy}^v \alpha_{vz}^u + \sum \alpha_{yz}^v \alpha_{vx}^u + \sum \alpha_{zx}^v \alpha_{vy}^u) u
    \end{align*}
\end{document}

however, i would write this equations as follows:

(red lines indicate text borders)

\documentclass{article}
\usepackage{mathtools} % <--- instead of `amsmath`, it define `multlined` environment

\begin{document}
    \begin{align*}
(f_{xy},f_{yz}) _{[w]}&=[f_{xy},z] - [x, f_{yz}]\\
     & = \Bigl[[x,y]- \sum \alpha_{xy}^v v ,z \Bigr]
        - \Bigl[x ,[y,z] -\sum\alpha_{yz}^v v \Bigr]\\
     & = [x,y,z] - \sum \alpha_{xy}^v [v,z] - \bigl[x,[y,z]\bigr] + \sum \alpha_{yz}^v [x,v]
\intertext{Using Jacobi's identity this equals:}
(f_{xy},f_{yz})_{[w]}
    & = \begin{multlined}[t][0.7\linewidth]
        \bigl[[x,z],y\bigr] - \sum\alpha_{xy}^v \Bigl([v,z] - \sum \alpha_{vz}^u u\Bigr)\\
        + \sum \alpha_{yz}^v 
        \Bigl( [x,v] - \sum \alpha_{xv}^u u \Bigr) - \sum \alpha_{xy}^v \alpha_{vz}^u u + \sum     \alpha_{yz}^v \alpha_{xv}^u u
        \end{multlined}    \\
    & = \begin{multlined}[t][0.7\linewidth]
        \Bigl[[x,z] - \sum \alpha_{xz}^v v , y \Bigr] \\
            + \sum \alpha_{xz}^v \left([v,y] - \sum \alpha_{vy}^u u\right) %\\
        -\sum \alpha_{xy}^v f_{vz} + \sum \alpha_{yz}^v f_{xv}  \\
        -\sum_{u} \Bigl(\sum \alpha_{xy}^v \alpha_{vz}^u + \sum \alpha_{yz}^v \alpha_{vx}^u + \sum \alpha_{zx}^v \alpha_{vy}^u\Bigr) u
        \end{multlined}     
    \end{align*}
\end{document}

Answer 2

A variant, with the multlined environment from mathtools, and the optional spacing argument of \intertext (defined in nccmath), and some delimiters size improvements:

 \documentclass{article}
\usepackage{mathtools, nccmath}

\begin{document}

    \begin{align*}
(f_{xy},f_{yz}) _{[w]}&=[f_{xy},z] - [x, f_{yz}]\\
         &=\bigl[[x,y]- \sum \alpha_{xy}^v v ,z \bigr] - \bigl[x ,[y,z] -\sum\alpha_{yz}^v v \bigr]\\
         &=\bigl[[x,y,z]\bigr]- \sum \alpha_{xy}^v [v,z] - \bigl[x,[y,z]\bigr] + \sum \alpha_{yz}^v [x,v]
\intertext[0.5ex]{Using Jacobi's identity this equals:}
 &=\begin{multlined}[t]{}
 \bigl[[x,z],y\bigr] - \sum\alpha_{xy}^v ([v,z] - \sum \alpha_{vz}^u u)\\
 + \sum \alpha_{yz}^v\bigl( [x,v] - \sum \alpha_{xv}^u u \bigr) - \sum \alpha_{xy}^v \alpha_{vz}^u u + \sum \alpha_{yz}^v \alpha_{xv}^u u
 \end{multlined}\\
 &= \begin{multlined}[t]{}%
 \bigl[[x,z] - \sum \alpha_{xz}^v v , y \bigr] + \sum \alpha_{xz}^v \bigl([v,y] - \sum \alpha_{vy}^u u\bigr)\\
 -\sum \alpha_{xy}^v f_{vz} + \sum \alpha_{yz}^v f_{xv}\\
 -\sum_{u} \Bigl(\sum \alpha_{xy}^v \alpha_{vz}^u + \sum \alpha_{yz}^v \alpha_{vx}^u + \sum \alpha_{zx}^v \alpha_{vy}^u\Bigr) u
 \end{multlined}
    \end{align*}

\end{document}