  The de Bruijn-Tengbergen-Kruyswijk (BTK) construction is a simple algorithm
that produces an explicit symmetric chain decomposition of a product of chains.
We linearize the BTK algorithm and show that it produces an explicit symmetric
Jordan basis (SJB). In the special case of a Boolean algebra the resulting SJB
is orthogonal with respect to the standard inner product and, moreover, we can
write down an explicit formula for the ratio of the lengths of the successive
vectors in these chains (i.e., the singular values). This yields a new,
constructive proof of the explicit block diagonalization of the Terwilliger
algebra of the binary Hamming scheme. We also give a representation theoretic
characterization of this basis that explains its orthogonality, namely, that it
is the canonically defined (upto scalars) symmetric Gelfand-Tsetlin basis.
