Checking antipode on Enveloping algebra of a Lie Algebra

Question

Let $U\left(\mathfrak{g}\right)$ be the enveloping algebra of $\mathfrak{g}$.Let's define comultiplication, counit and antipode as $$\triangle\left(X\right) =X\otimes1+1\otimes X,$$ $$\epsilon\left(X\right) =0,$$ $$S\left(X\right) =-X.$$ We can read everywhere that $U\left(\mathfrak{g}\right)$ with those applications is an hopf algebra. I'm checking the antipode condition and I find $$ \mu\circ\left(S\otimes id\right)\circ\triangle\left(X\right)=$$ $$=\mu\circ\left(S\otimes id\right)\left(X\otimes1+1\otimes X\right)=$$ $$=\mu\circ\left(-X\otimes1+-1\otimes X\right)=$$ $$=-X-X=-2X$$ While it should be zero... What did I do wrong?

Answer 1

The extension of the antipode map to $\mathcal{U}(\mathfrak{g})$ maps each scalar to itself. Therefore$$(S\otimes\operatorname{id})(X\otimes1+1\otimes X)=-X\otimes1+1\otimes X$$and $\mu(-X\otimes1+1\otimes X)=0$.