Why can't split with curly-brace be inside align but it can be inside gather?

Question

The following code snippet will guide you through the issue chronologically.

\documentclass[preview,border=12pt,12pt]{standalone}
\usepackage{mathtools}
\begin{document}
\section*{gather+aligned}
\begin{gather}
    \left\{\!
    \begin{aligned}
        x &= \cos t\\
        y &= \sin t
    \end{aligned}\right.\\
    x^2 +y^2 =1
\end{gather}


\section*{gather+split}
\begin{gather}
    \left\{
    \begin{split}
        x &= \cos t\\
        y &= \sin t
    \end{split}\right.\\
    x^2 +y^2 =1
\end{gather}


\section*{align+aligned}
\begin{align}
    &\left\{\!
    \begin{aligned}
        x &= \cos t\\
        y &= \sin t
    \end{aligned}\right.\\
    &x^2 +y^2 =1
\end{align}


\section*{align+split}
It cannot be compiled.
%\begin{align}
    %&\left\{
    %\begin{split}
        %x &= \cos t\\
        %y &= \sin t
    %\end{split}\right.\\
    %&x^2 +y^2 =1
%\end{align}

\section*{align+split+nobrace}
It can be compiled.
\begin{align}
    &
    \begin{split}
        x &= \cos t\\
        y &= \sin t
    \end{split}\\
    &x^2 +y^2 =1
\end{align}

\end{document}

Why can't split with curly-brace \left\{ and \right. be inside align but it can be inside gather?

Answer 1

From the amsmath user guide:

The split environment is designed to serve as the entire body of an equation, or an entire line of an align or gather environment. There cannot be any printed material before or after it within the same enclosing structure.

In particular, you can't have \left\{ to the left and \right. to the right of the split material.

For more on this subject, see the posting What's the difference between split and aligned?