  I construct classical superextensions of the Virasoro algebra by employing
the Ward identities of a linearly realized subalgebra. For the $N=4$
superconformal algebra, this subalgebra is generated by the $N=2$ $U(1)$
supercurrent and a spin~0 $N=2$ superfield. I show that this structure can be
extended to an $N=4$ super $W_3$ algebra, and give the complete form of this
algebra.
