  Let there be given a probability measure $\mu$ on the unit circle $\TT$ of
the complex plane and consider the inner product induced by $\mu$. In this
paper we consider the problem of orthogonalizing a sequence of monomials
$\{z^{r_k}\}_k$, for a certain order of the $r_k\in\mathbb{Z}$, by means of the
Gram-Schmidt orthogonalization process. This leads to a basis of orthonormal
Laurent polynomials $\{\psi_k\}_k$. We show that the matrix representation with
respect to the basis $\{\psi_k\}_k$ of the operator of multiplication by $z$ is
an infinite unitary or isometric matrix allowing a 'snake-shaped' matrix
factorization. Here the 'snake shape' of the factorization is to be understood
in terms of its graphical representation via sequences of little line segments,
following an earlier work of Delvaux and Van Barel. We show that the shape of
the snake is determined by the order in which the monomials $\{z^{r_k}\}_k$ are
orthogonalized, while the 'segments' of the snake are canonically determined in
terms of the Schur parameters for $\mu$. Isometric Hessenberg matrices and
unitary five-diagonal matrices (CMV matrices) follow as a special case of the
presented formalism.
