In notes:

x1 is a binary group operator endowing set/space x2 ; x2 is the underlying set or the actual structure of a group with operator x1.
x1 is a binary operator which is commutative in space/under conditions/on (or endowing) set x2; x1 and x2 are each abelian (in different senses).
x1 is a binary operator in space/under conditions/on (or endowing) set x2 such that there is an identity element and for any element for which it is defined in the space (excepting possibly a small number of special elements), there exists at least one element also in the space which is a left-inverse of that element under the operator.
x1 is an n-ary operator/map which is distributive/linear/homomorphic in or over or from space/structure x2, mapping to space or structure x3, thereby producing a new space/structure x4 which is the 'union' of x2 and x3 endowed with x1; x1 distributes over/through all of the operators of x2.
x1 is a binary operator in structure x2 which exhibits the Jacobi property with respect to binary operator x3 (which also endows x2) and element/object x4 (which is an element of the underlying set which form x2).