The Lie algebra does not detect nonsmooth group schemes

The subgroup scheme ${\alpha }_p$ is not smooth at its origin. Its coordinate ring is $k[s]\mathrm{/}(s^p)$ with $s\mathrm{\ne }0$ and $s^p=0$ by [F1], so it is not reduced. Every element outside $(s)$ is a unit by a finite geometric sum, so localization at $(s)$ leaves this ring unchanged and the class $s$ remains nonzero. If ${\alpha }_p$ were smooth at the origin, [F3] with the trivial extension $k\mathrm{/}k$ would make the local ring $k[s]\mathrm{/}(s^p{)}_{(s)}$ regular, and [F4] would make that ring an integral domain, contradicting $s\mathrm{\ne }0$ with $s^p=0$; hence ${\alpha }_p$ is not smooth over $k$. The same argument with the same coordinate ring, using $s=t-1$, shows that ${\mu }_p$ is not smooth over $k$.

The ambient groups are smooth. The polynomial ring $k[t]$ is a localisation of a polynomial ring and its Jacobian presentation has no equations, so it is standard smooth over $k$; by [F5] the morphism ${\mathbf{G}}_a\to \mathrm{Spec}k$ is flat with geometrically regular fibres and locally of finite presentation over the Noetherian field $k$, hence smooth by definition. The same presentation with no equations applies to $k[t,t^{-1}]=(k[t]{)}_t$, so ${\mathbf{G}}_m$ is smooth over $k$.

The Lie algebras agree. By [F2] there are isomorphisms of $k$-Lie algebras $\mathrm{Lie}({\alpha }_p)\cong \mathrm{Lie}({\mathbf{G}}_a)$ and $\mathrm{Lie}({\mu }_p)\cong \mathrm{Lie}({\mathbf{G}}_m)$. Combining this with steps 1.1 and 1.2, the nonsmooth finite group scheme ${\alpha }_p$ and the smooth group scheme ${\mathbf{G}}_a$ have isomorphic Lie algebras, and likewise for ${\mu }_p$ and ${\mathbf{G}}_m$; therefore the Lie algebra of a group scheme of finite type over a field does not determine smoothness, and the refuted statement fails. Choice is inherited through the matrix-group and Lie-bracket suppliers [F1, F2], the regular-local-domain supplier [F4], and the standard-smoothness criterion [F5].