label equations in cases environment inside align environment

Question

I want to label all equations, including those in cases environment, which is inside align environment. Here is my example:

\begin{align} 
\delta_j \delta_i &= \delta_i \delta_{j-1} \text{\quad for }i<j, \\ 
 \sigma_j\sigma_i&=\sigma_i\sigma_{j+1} \text{\quad for } i \leq j,  \\ 
\sigma_j\delta_i&= \begin{cases}
\delta_i \sigma_{j-1} & \text{for } i<j, \\
\id_{\left[n\right]} & \text{for } i=j,~~ i=j+1,\\
\delta_{i-1}\sigma_j & \text{for } i>j+1.
\end{cases}
\end{align}

After compiling, the part coming with the cases environment is labeled as a single equation, but I want to label each case. Would it be possible while keeping the align environment.

Answer 1

Here is a hack with alignat:

\documentclass{article}
\usepackage{mathtools, amssymb}
\DeclareMathOperator{\id}{id}

\begin{document}

\begin{alignat}{3}
\delta_j \delta_i &=\mathrlap{\delta_i \delta_{j-1}} & & & \quad & \text{for }i<j, \\
 \sigma_j\sigma_i&=\mathrlap{\sigma_i\sigma_{j+1}} & & & \quad & \text{for } i \leq j, \\
 & & & \delta_i \sigma_{j-1} & \quad & \text{for } i<j, \\
\sigma_j\delta_i & = &\smash{\begin{cases}{}\\{} \\ {}\end{cases}}\hskip-1.2em & \id_{\left[n\right]} & & \text{for } i=j,~~ i=j+1,\\
 & & & \delta_{i-1}\sigma_j & & \text{for } i>j+1.
\end{alignat}

\end{document}