This identity is a special case of Euler's Identity for Continuants. It is a Pfaffian of degenerate $4\times 4$ matrix.

As Michael Somos mentioned in his comment it is a part of "elliptic realm" where different identities arise as determinants of degenerate matrices. These matrices are degenerate because they are submatrices of infinite matrices of finite rank. For examle the matrix with entries $a_{m,n}=s_{m+n}s_{m-n}$ ($m,n\in \mathbb{Z}$) where $s_n$ is the Somos-4 seqence has rank $2$. For Somos-6 corresponding matrix has rank $4$ etc.

This identity is a special case of Euler's Identity for Continuants. It is a Pfaffian of degenerate $4\times 4$ matrix. Concrete mathematics gives the following reference:

As Michael Somos mentioned in his comment it is a part of "elliptic realm" where different identities arise as determinants of degenerate matrices. These matrices are degenerate because they are submatrices of infinite matrices of finite rank. For examle the matrix with entries $a_{m,n}=s_{m+n}s_{m-n}$ $(m,n\in \mathbb{Z})$ where $s_n$ is the Somos-$4$ seqence has rank $2$. For Somos-$6$ corresponding matrix has rank $4$ etc.