Center of the infinite dihedral group algebra ($Z(\mathbb{C}[D_{\infty }])=\mathbb{C}[X+X^{-1}]$)

Write the group algebra as $\mathbb{C}[X,X^{-1}]\oplus \mathbb{C}[X,X^{-1}]Y$, with $Y^2=1$ and $YX=X^{-1}Y$. Commuting $f(X)+g(X)Y$ with $X$ forces $g=0$, and commuting with $Y$ forces $f(X)=f(X^{-1})$. Every symmetric Laurent polynomial is a polynomial in $z=X+X^{-1}$, using the recursion $X^{m+1}+X^{-m-1}=z(X^m+X^{-m})-(X^{m-1}+X^{1-m})$. The algebra is free of rank four over this center, with basis $1,X,Y,XY$; the relation $X^2-zX+1=0$ proves spanning, and applying $X\mapsto X^{-1}$ proves independence of $1,X$.