= Solution
The addition formulas for the <hyperbolic functions> imply
$$
B(u)B(v)=B(u+v),\qquad B(u)=\begin{pmatrix}\cosh u&\sinh u\\\sinh u&\cosh u\end{pmatrix}.
$$
Since $M=B(y)/2$, its fourth <matrix power> is
$$
\boxed{M^4=\frac1{16}\begin{pmatrix}\cosh4y&\sinh4y\\\sinh4y&\cosh4y\end{pmatrix}.}
$$
Alternatively, the orthonormal <eigenvectors> $(1,\pm1)/\sqrt2$ have <eigenvalues> $e^{\pm y}/2$. Raising these <eigenvalues> to the fourth power gives the same result by <spectral decomposition>.
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