Definitions

A loop is a set L with a binary operation  such that 1) for each a,b in L, there is a unique x in L such that a  x=b and there is a unique y in L such that y  a=b. 2) there exists a unique element 1 in L such that for every a in L, 1  a=a=a  1. We will simply write ab in place of a  b. A (right) Bol loop is a loop L in which the identity [(xy)z]y = x[(yz)y] holds for all x,y,z in L.

The center Z(L) of a loop L is the set of elements z in L such that for all a, b in L, a(zb)=(az)b=(za)b=z(ab)=(ab)z=a(bz) The center of a loop is a normal subloop (the kernel of a loop homomorphism). A loop is centrally nilpotent of class n if the upper central series of L {1}  Z=Z 1  Z 2  …  Z n  …, where Z i+1 is the full preimage in L of the center of L/Z i, stabilizes with Z n =L, but Z n-1  L.

For a, b, c in L, The commutator (a,b) is defined by ab=(ba)(a,b) The associator (a,b,c) is defined by (ab)c=[a(bc)](a,b,c) The commutator/associator subloop L is the subloop of L generated by all commutators and all associators.

If L is of nilpotence class 2, then Z 2 =L, so that L/Z(L) is an abelian group. Therefore Z(L) contains all commutators and all associators. If z  Z(L), then (za,b)=(a,zb)=(ab) (za,b,c)=(a,zb,c)=(a,b,zc)=(a,b,c) Also commutators and associators can be pulled to the right in all equations. E.g., [x(x,y)][y(x,y,z)]=(xy)(x,y)(x,y,z)