If $V$ is a vector space over a field $K$, then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\in V$. The homogeneous component of degree $n$ of $S(V)$ is $S^n(V)=T^n(V)/I_n$, where $I_n$ is th...