Let $V$ be an finite-dimensional vector space, let $\alpha$ be a symmetric $k-$tensor on $V$, and let $\beta$ be a symmetric $l-$tensor on $V$. Define the symmetric product of $\alpha$ and $\beta$ to be the $\left(k+l\right)$-tensor $\alpha\beta$ defined by $\left(\alpha\beta\right)\left(v_{1},\ldots,v_{k+l}\right)=\frac{1}{\left(k+l\right)!}\sum_{\sigma\in S_{k+l}}\alpha\left(v_{\sigma(1)},\ldots,v_{\sigma(k)}\right)\beta\left(v_{\sigma(k+1)},\ldots,v_{\sigma(k+l)}\right)$ for all $v_{1},\ldots,v_{k+l}\in V$.
Show that $\alpha\beta=\beta\alpha$, that is,
$$\sum_{\sigma\in S_{k+l}}\alpha\left(v_{\sigma(1)},\ldots,v_{\sigma(k)}\right)\beta\left(v_{\sigma(k+1)},\ldots,v_{\sigma(k+l)}\right)=\sum_{\sigma\in S_{k+l}}\beta\left(v_{\sigma(1)},\ldots,v_{\sigma(l)}\right)\alpha\left(v_{\sigma(l+1)},\ldots,v_{\sigma(l+k)}\right)$$.
for all $v_{1},\ldots,v_{k+l}\in V$.