& 0      &         &        &           &          &           & \\
0 & 0      & \beta_{2,3}  &   &           &          &           & \\
  & -\beta_{2,3}& 0       & 0 &           &          &           & \\
  &        & 0       & \ddots & 0         &          &           & \\
  &        &         & 0      & 0         & \beta_{n-3,n-2} &     & \\
  &        &         &        & -\beta_{n-3,n-2} & 0        & 0   & \\
  &        &         &        &           & 0        & 0  & \beta_{n-1,n} \\
  &        &         &        &           &          & -\beta_{n-1,n} & 0
\end{array} \right).
\label{h2m}$$ Clearly both $H_{1}$ and $H_{2}$ are skew-symmetric block-diagonal matrices, containing one $1\times1$ matrix and $(n-1)/2$ real, $2\times2$ skew-symmetric matrices.

According to the general theory given above, the first-order algorithm defined by $$U_1(\tau)=e^{\tau H_{1}} e^{\tau H_{2}}.
\label{U1}$$ is all that is needed to construct unconditionally stable second and higher-order algorithms. As the matrix exponential of a block-diagonal matrix is equal to the block-diagonal matrix of the matrix exponentials of the individual blocks, the numerical calculation of $e^{\tau H_{1}}$ (or $e^{\tau H_{2}}$) reduces to the calculation of $(n-1)/2$ matrix exponentials of $2\times2$ matrices. The matrix exponential of a typical $2\times2$ matrix appearing in $e^{\tau H_{1}}$ or $e^{\tau H_{2}}$ is given by $$\begin{aligned}
\exp\left[\alpha
\left(\begin{array}{cc} {\phantom{+}}0&1\\ -1&0\end{array}\right)\right]
