Universal enveloping algebra of sl2

Question

I am currently trying to prove that $x-1$ is not invertible in the universal enveloping algebra $\mathfrak{U}(\mathfrak{sl}(2,\mathbb{F}))$ of $\mathfrak{sl}(2, \mathbb{F})$, but I still struggle with understanding how $\mathfrak{U}(\mathfrak{sl}(2,\mathbb{F}))$ itself looks.
Let $x=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, $h=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ and $y=\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ denote the standard basis of $\mathfrak{sl}(2, \mathbb{F})$. I tried working with the corresponding PBW basis. First I took an arbitrary element $u=\sum\nolimits_{ijk}\lambda_{ijk}x^{i}h^{j}y^{k}$ with $(x-1)u=1$ and tried to show that such an $u$ doesn´t exist.
\begin{align*}1&=(x-1)u \\ &=(x-1)(\sum\nolimits_{ijk}\lambda_{ijk}x^{i}h^{j}y^{k}) \\ &=(\sum\nolimits_{ijk}\lambda_{ijk}x^{i+1}h^{j}y^{k})-(\sum\nolimits_{ijk}\lambda_{ijk}x^{i}h^{j}y^{k}) \\ &= \dots\end{align*} But I am not even sure if this is the correct way working in $\mathfrak{U}(\mathfrak{sl}(2,\mathbb{F}))$.
Thank you for helping me.