It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative.

For if a and b are elements of R, and writing + for the group operation then applying the distributive property one has

$a+a+b+b=a*(1+1)+b*(1+1)=(a+b)*(1+1)= (a+b)*1+(a+b)*1=a+b+a+b$, whence $a+b=b+a$.

For educational purposes, are there more (not only algebraic) examples of such superfluous definitions?

It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative.

For if a and b are elements of R, and writing + for the group operation then applying the distributive property one has $$ \begin{align} a+a+b+b&=a*(1+1)+b*(1+1)\\\\ &=(a+b)*(1+1)\\\\ &=(a+b)*1+(a+b)*1\\\\ &=a+b+a+b, \end{align} $$ whence $a+b=b+a$.

For educational purposes, are there more (not only algebraic) examples of such superfluous definitions?