Conservative algebras of 2-dimensional algebras

In 1990 Kantor introduced the conservative algebra W(n)W(n) of all algebras (i.e. bilinear maps) on the n  -dimensional vector space. In case n>1n>1 the algebra W(n)W(n) does not belong to well-known classes of algebras (such as associative, Lie, Jordan, Leibniz algebras). We describe the algebra of all derivations of W(2)W(2) and subalgebras of W(2)W(2) of codimension one. We also study similar problems for the algebra W2W2 of all commutative algebras on the two-dimensional vector space and the algebra S2S2 of all commutative algebras with trace zero multiplication on the two-dimensional space.