I saw somewhere (I appreciate if anyone has any references to proof of this fact) that if $A$ is a finite dimensional associative algebra such that $\textrm{dim}(A)<n$, then $A$ satisfies the standard identity of degree $n$: $$s_n(x_1,\dots, x_n)=\sum\limits_{\sigma\in S_n}\textrm{sgn}(\sigma)x_{\sigma(1)}\cdots x_{\sigma(n)},$$ where $S_n$ is the symmetric group of degree $n$.

Is there a non-associative version of this result? I mean, if $A$ is a finite-dimensional (not necessarily associative) algebra, then $A$ satisfies some kind of "non-associative" standard polynomials? For example, maybe: $$P_1(x_1)=x_1=s_1(x_1),$$ $$P_2(x_1,x_2)=x_1x_2-x_2x_1=s_2(x_1,x_2),$$ $$P_3(x_1,x_2,x_3)=x_1(x_2x_3)-(x_1x_2)x_3-x_1(x_3x_2)+(x_1x_3)x_2-x_2(x_1x_3)+(x_2x_1)x_3+x_2(x_3x_1)-(x_2x_3)x_1+x_3(x_1x_2)-(x_3x_1)x_2-x_3(x_2x_1)+(x_3x_2)x_1=x_1(x_2x_3-x_3x_2)+x_2(x_3x_1-x_1x_3)+x_3(x_1x_2-x_2x_1)+(x_3x_2-x_2x_3)x_1+(x_1x_3-x_3x_1)x_2+(x_2x_1-x_1x_2)x_3$$ and etc... Thanks in advance for answers!