  We analyze the W_N^l algebras according to their conjectured realization as
the second Hamiltonian structure of the integrable hierarchy resulting from the
interchange of x and t in the l^{th} flow of the sl(N) KdV hierarchy. The W_4^3
algebra is derived explicitly along these lines, thus providing further support
for the conjecture. This algebra is found to be equivalent to that obtained by
the method of Hamiltonian reduction. Furthermore, its twisted version
reproduces the algebra associated to a certain non-principal embedding of sl(2)
into sl(4), or equivalently, the u(2) quasi-superconformal algebra. The general
aspects of the W_N^l algebras are also presented.
