Dendriform algebra is a well-known non-associative algebra defined by Loday and Ronco. More precisely,
A Dendriform algebra is a $k$-module $D$ together with two binary operations $\{\prec, \succ\}$, such that, \begin{align} (x\prec y) \prec z=\ & x \prec (y\prec z) +x\prec (y \succ z), \\ % (x\succ y)\prec z=\ & x\succ (y\prec z),\quad \quad \quad \quad \quad \ \ \ \ \\ % x\succ(y \succ z)=\ & (x\prec y)\succ z +(x\succ y) \succ z， \end{align} for $x, y, z\in D$. By the second identity, it looks not symmetric, can we add $(x\prec y)\succ z=\ x\prec (y\succ z)$? Or under some condition, commutative? can we get $(x\prec y)\succ z=\ x\prec (y\succ z)$?