Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)In 1990 Kantor introduced the conservative algebra W(n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. In case n > 1 the algebra 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) and subalgebras of W(2) of codimension one. We also study similar problems for the algebra W-2 of all commutative algebras on the two-dimensional vector space and the algebra S-2 of all commutative algebras with trace zero multiplication on the two-dimensional space. (C) 2015 Elsevier Inc. All rights reserved.486 255274Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)RFBR [14-01-31084, 14-01-31122]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)