A group is a set with a distinguished element e and a binary operation [ , ] satisfying: ‣ [a,b] = [[a,c],[b,c]] ‣ [a,e] = a ‣ [a,a] = e for all a,b,c.
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Form a ring with one more operator.
I suspect there's nothing interesting to do for the ring multiplication, because this definition really relies on having inverses. But one thing you'd want to do anyway is express the notion of "abelian group". The best I can do is ‣ [a,b]=[[e,b],[e,a]]
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This encodes the commutative property, but is there a nicer way to do so?
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What about [a, [b, c]] = [c, [b, a]]
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