For an isometry $A$ on Hilbert space $H$, the task of an efficient
orthogonalization of the Krylov sequence $x$, $Ax$, $A^2x$,
$\ldots$ (with additional specific subtractions admitted) is
addressed and solved in a matrix generalization where $x \to M_j$
becomes a sequence of $p-$plets of elements of $H$. Several
samples of applicability (like the study of structured Toeplitz
matrices, of the role of circulants or of a reduction of $A$ to a
product of plane rotations etc) are mentioned. Connections with
the Gram-Schmidt approach, $QR$ factorization technique and
generalized Schur algorithm (with displacements) are discussed and
a few encouraging numerical comparisons are added.

MR2312401 Stewart, Michael A generalized iosmetric Arnoldi algorithm.
Linear Algebra Appl. 423 (2007), no. 2-3, 183--208. 65F15