flexible algebra

A non-associative algebra $A$ is flexible if $[a,b,a]=0$ for all $a,b\in A$, where $[,,]$ is the associator on $A$. In other words, we have $(ab)a=a(ba)$ for all $a,b\in A$. Any associative algebra is clearly flexible. Furthermore, any alternative algebra with characteristic $\ne 2$ is flexible.

Given an element $a$ in a flexible algebra $A$, define the left power of $a$ iteratively as follows:

1. $L^1(a)=a$,

2. $L^n(a)=a\cdot L^{n-1}(a)$.

Similarly, we can define the right power of $a$ as:

1. $R^1(a)=a$,

2. $R^n(a)=R^{n-1}(a)\cdot a$.

Then, we can show that $L^n(a)=R^n(a)$ for all positive integers $n$. As a result, in a flexible algebra, one can define the (multiplicative) power of an element $a$ as $a^n$ unambiguously.