  In 1990 Kantor defined the conservative algebra $W(n)$ of all algebras (i.e.
bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra
$W(n)$ does not belong to any well-known class of algebras (such as
associative, Lie, Jordan, or Leibniz algebras). We describe automorphisms,
one-sided ideals, and idempotents of $W(2).$ Also similar problems are solved
for the algebra $W_2$ of all commutative algebras on the 2-dimensional vector
space and for the algebra $S_2$ of all commutative algebras with trace zero
multiplication on the 2-dimensional vector space.
