Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients $(a_0, \dotsc, a_n) $ for a basis $\{1, i_1, \dotsc, i_n \}$.
If $a,b,c,d$ are invertible elements of the algebra then $ab = c$ and $ad = c$ implies $b = d$. (cancellation property)
This implies the answer can be given in the form of a Cayley table for multiplying the finite dimensional basis $\{1, i_1, \dotsc, i_n \}$.
So our algebra is of the form $ a_0 + a_1 i_1 + a_2 i_2 + \dotsb$, addition is as usual and products are defined by a Cayley table relating the basis elements $\{1, i_1, \dotsc, i_n \}$ And clearly the Cayley table is a latin square.
Assume the algebra is also power-associative.
One important extra condition:
Let $T$ be such an Algebra.
Let $x$ be an element of $T$. Then $y^2 = x$ always has at least one solution $y$ within $T$.
So basically the square root is defined in the algebra.
We also need "Property A"
$$ x * (y * y^{-1}) = x = (x * y) * y^{-1} $$
holds for all invertible elements $x,y$ in the algebra.
( if this property has a name please tell me )
The final condition is that this type of algebra $T$ we seek, is not isomorphic to a tensor product of the complex numbers with another algebra $B$ (taken over the reals).
The dimensions of such an algebra are restricted and must be of the form " extended Mersenne number " +1 or a subset of them.
These extented Mersenne numbers were already discussed before :
Consider the subset of odd positive integers defined and constructed as follows by these rules :
A) $1$ is in the set.
B) if $x$ is in the set , then $2x + 1$ is in the set.
C) if $x$ and $y$ are in the set then $xy$ is in the set.
I call them extended Mersenne numbers because rule A and B alone give the Mersenne numbers $2^n - 1$.
And every product of Mersenne numbers must be in the set as well.
So the set or list of extended Mersenne numbers starts like
$$1,3,7,9,15,19,21,27,31,39,43,45,49,55,57,63,79,81,87,91,93,99,...$$
see here :
Density of extended Mersenne numbers?
Now the main question is this :
When does such a commutative non-associative algebra have nilpotent elements ??
Does it always have nilpotent elements ?
Or never ? sometimes ?
examples ?
