I need some clarification on the definition of free algebra. Here is an extract from Lie Algebras and Lie Groups by Jean-Pierre Serre:
I am somewhat confused about the definition of free algebra. Let say $X=\{x\}$ and $k=\mathbb{R}$, then the free algebra $A_X$ consists of all elements of the form $$rx,rxx, r((xx)x), r(x(xx)), r(((xx)x)x), r((x(xx))x), \cdots (r\in\mathbb{R}),$$ right? Am I right in saying that in $A_X$, $(xx)x$ and $x(xx)$ are different elements?