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, \dots, a_n) $ for a basis $\{1, i_1, \dots, i_n \}$ and we have a basis so that $ i_k^2 \in \{ -1, +1 \}$.
This implies the answer can be given in the form of a cayley table for multiplying the finite dimensional basis $\{1, i_1, \dots, i_n \}$.
So our algebra is of the form $ a_0 + a_1 i_1 + a_2 i_2 + ...$, addition is as usual and products are defined by a cayley table relating the basis elements $\{1, i_1, \dots, i_n \}$ and we have a basis so that $ i_k^2 \in \{ -1, +1 \}$.
Assume the basis elements are also power-associative.
Does this imply the whole algebra is power-associative and nilpotent elements do not exist ??