Suppose $\mathfrak{g}$ is a semisimple Lie algebra, with enveloping algebra $\mathcal{U}(\mathfrak{g})$. Then in the standard PBW decomposition, $\mathcal{U}(\mathfrak{g}) \cong \mathcal{U}(\mathfrak{n}^-) \otimes \mathcal{U}(\mathfrak{h}) \otimes \mathcal{U}(\mathfrak{n}^+)$. Then we may choose an ordering of the positive roots $\alpha_1, \ldots, \alpha_n$ say and denote the "standard PBW ordering" to be of the form, $$y_1^{r_1}\ldots y_m^{r_m}h_1^{s_1}\ldots h_l^{s_l} x_1^{t_1} \ldots x_m^{t_m},$$ when split into $\mathfrak{sl}_{\alpha}$ triples.
My question concerns putting elements into "standard order" of the form above. For example take $\mathfrak{sl}_3(\mathbb{C})$, with positive roots $\alpha, \beta, \gamma = \alpha + \beta$. Then when expressing a term such as $x_{\alpha}y_\alpha y_\beta$ in the standard order we write, $$x_{\alpha}y_\alpha y_\beta = y_\alpha y_\beta x_\alpha + y_\beta h_\alpha + y_\beta. $$
Given that the term on the left is zero, and that the first term on the right is zero too (and in the "standard order") what is the purpose of the second and third terms on the right, which cancel ($y_\beta h_\alpha = - y_\beta$)?
I suspect I may be missing something obvious here but any information would be appreciated.