Complex and real generalized Schur forms

Let $(A,B)$ be a regular pencil of order $n$. A complex generalized Schur form is a factorisation $Q^{\ast }AZ=S,Q^{\ast }BZ=T,$ where $Q,Z\in {\mathbb{C}}^{n\times n}$ are unitary in the sense of Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces, and $S,T$ are upper triangular as in Upper triangular, lower triangular and diagonal square matrices over a commutative ring. For its $i$th one-by-one diagonal block, $[s_{ii}:t_{ii}]$ is the diagonal homogeneous pair.

For real $A,B$, a real generalized Schur form has $Q^{T}AZ=S,Q^{T}BZ=T,$ where $Q,Z$ are orthogonal, $S$ is block upper triangular with diagonal blocks of size one or two, and $T$ is upper triangular. Each one-by-one diagonal block has a nonzero real homogeneous pair. For a two-by-two diagonal block $(S_{kk},T_{kk})$, the diagonal homogeneous roots are the zeros of $\det (\beta S_{kk}-\alpha T_{kk})$ in $[\alpha :\beta ]$; such a block has a nonreal conjugate pair of finite roots. A singular pencil is outside this definition.