Let A be the mod 2 Steenrod algebra and H**(A) = Exti*(Z2, Z2) its cohomology. The ultimate aim in studying H**(A) is the long-standing problem of computing the homotopy groups of spheres via the Adams spectral sequence £1]. H(A) has been computed up to certain values of t — s by Adams £4], Ivanovskii £6] (see also: International Congress of Mathematicians, Moscow 1966), Liulevicius £8], May £12, 13], Tangora £21]. It is of interest to know any systematic phenomena in H**(A). In this direction a polynomial wedge subalgebra of H**(A) has been obtained by Mahowald and Tangora £9]. Also: Margolis, Priddy and Tangora proved in [[10] that the Mahowald-Tangora wedge subalgebra is repeated every 45 stems, under the action of a specific periodicity operator. The present writer has described shortly in £24] a polynomial subalgebra of J3**(^4) generated by J0, eQ, g. This subalgebra will be described here in more detail. The basic technique is to study H**(A) by studying H**(B) for a suitable subalgebra B of A. This technique is due to Adams £3]. It has also been used by Margolis, Priddy and Tangora £10]. Moreover G.W. Whitehead £22] shows that, using the Adams technique, one can obtain many polynomial subalgebras of H**(A). The present paper is organized as follows: In Section 2 we state the main theorem and sketch its proof. The detailed proof involves the Adams technique and the construction of the generators o?0, e0, g. These constructions use known relations between the classes h{ (see Adams £2], Novikov £19]); they also use Steenrod U rproducts in F(A*). In Section 3 we describe briefly these cup-i-products and in Section 4 we give the