Consider a non-associative commutative unital algebra of finite dimension 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)
Assume the algebra is also power-associative and NOT nilpotent.
Can we always pick a basis, such that the multiplication table (for the basis elements) is a loop without having products equal to "sums ($+$)" of other basis elements ?