From the amsmath documentation:
The aligned environment takes an optional argument that indicates its vertical position in relation to surrounding material: t, c, or b for top, center, or bottom.
In your code this optional argument is read as [A_i, A_j], which is unknown. And you get the warning reported in the other answer by @daleif.
Thus you have to write
\documentclass[12pt,a4paper]{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}\text{at top }
\begin{aligned}[t]
[A_i, A_j]&=i\varepsilon_{ijk}A_k\\
[B_i, B_j]&=i\varepsilon_{ijk}B_k\\
[A_i, B_j]&=0
\end{aligned}\text{ after eq}
\label{eq:10t}
\end{equation}
\begin{equation}\text{centered }
\begin{aligned}[c]
[A_i, A_j]&=i\varepsilon_{ijk}A_k\\
[B_i, B_j]&=i\varepsilon_{ijk}B_k\\
[A_i, B_j]&=0
\end{aligned}\text{ after eq}
\label{eq:10c}
\end{equation}
\begin{equation}\text{at bottom }
\begin{aligned}[b]
[A_i, A_j]&=i\varepsilon_{ijk}A_k\\
[B_i, B_j]&=i\varepsilon_{ijk}B_k\\
[A_i, B_j]&=0
\end{aligned}\text{ after eq}
\label{eq:10b}
\end{equation}
\end{document}
amsmathpackage. On my system (MacTeX2020, w/ all updates installed), your code does not cause[A_i, A_j]to disappear. – Mico Jan 12 '21 at 08:21