By Varieties, Lemma \ref{varieties-lemma-ample-after-field-extension} we may assume that $k$ is algebraically closed. Let $G^0 \subset G$ be the connected component of $G$ as in Proposition \ref{proposition-connected-component}. Then every other connected component of $G$ has a $k$-rational point and hence is isomorphic to $G^0$ as a scheme. Since $G$ is quasi-compact and Noetherian, there are finitely many of these connected components. Thus we reduce to the case discussed in the next paragraph. \medskip\noindent Let $G$ be a connected algebraic group scheme over an algebraically closed field $k$. If the characteristic of $k$ is zero, then $G$ is smooth over $k$ by Lemma \ref{lemma-group-scheme-characteristic-zero-smooth}. If the characteristic of $k$ is $p > 0$, then we let $H = G_{red}$ be the reduction of $G$. By Divisors, Proposition \ref{divisors-proposition-push-down-ample} it suffices to show that $H$ has an ample invertible sheaf. (For an algebraic scheme over $k$ having an ample invertible sheaf is equivalent to being quasi-projective over $k$, see for example the very general More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-projective}.) By Lemma \ref{lemma-reduced-subgroup-scheme-perfect} we see that $H$ is a group scheme over $k$. By Lemma \ref{lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth} we see that $H$ is smooth over $k$. This reduces us to the situation discussed in the next paragraph. \medskip\noindent Let $G$ be a quasi-compact irreducible smooth group scheme over an algebraically closed field $k$. Observe that the local rings of $G$ are regular and hence UFDs (Varieties, Lemma \ref{varieties-lemma-smooth-regular} and More on Algebra, Lemma \ref{more-algebra-lemma-regular-local-UFD}). The complement of a nonempty affine open of $G$ is the support of an effective Cartier divisor $D$. This follows from Divisors, Lemma \ref{divisors-lemma-complement-open-affine-effective-cartier-divisor}. (Observe that $G$ is separated by Lemma \ref{lemma-group-scheme-over-field-separated}.) We conclude there exists an effective Cartier divisor $D \subset G$ such that $G \setminus D$ is affine. We will use below that for any $n \geq 1$ and $g_1, \ldots, g_n \in G(k)$ the complement $G \setminus \bigcup D g_i$ is affine. Namely, it is the intersection of the affine opens $G \setminus Dg_i \cong G \setminus D$ in the separated scheme $G$. \medskip\noindent We may choose the top row of the diagram $$ \xymatrix{ G & U \ar[l]_j \ar[r]^\pi & \mathbf{A}^d_k \\ & W \ar[r]^{\pi'} \ar[u] & V \ar[u] } $$ such that $U \not = \emptyset$, $j : U \to G$ is an open immersion, and $\pi$ is \'etale, see Morphisms, Lemma \ref{morphisms-lemma-smooth-etale-over-affine-space}. There is a nonempty affine open $V \subset \mathbf{A}^d_k$ such that with $W = \pi^{-1}(V)$ the morphism $\pi' = \pi|_W : W \to V$ is finite \'etale. In particular $\pi'$ is finite locally free, say of degree $n$. Consider the effective Cartier divisor $$ \mathcal{D} = \{(g, w) \mid m(g, j(w)) \in D\} \subset G \times W $$ (This is the restriction to $G \times W$ of the pullback of $D \subset G$ under the flat morphism $m : G \times G \to G$.) Consider the closed subset\footnote{Using the material in Divisors, Section \ref{divisors-section-norms} we could take as effective Cartier divisor $E$ the norm of the effective Cartier divisor $\mathcal{D}$ along the finite locally free morphism $1 \times \pi'$ bypassing some of the arguments.} $T = (1 \times \pi')(\mathcal{D}) \subset G \times V$. Since $\pi'$ is finite locally free, every irreducible component of $T$ has codimension $1$ in $G \times V$. Since $G \times V$ is smooth over $k$ we conclude these components are effective Cartier divisors (Divisors, Lemma \ref{divisors-lemma-weil-divisor-is-cartier-UFD} and lemmas cited above) and hence $T$ is the support of an effective Cartier divisor $E$ in $G \times V$. If $v \in V(k)$, then $(\pi')^{-1}(v) = \{w_1, \ldots, w_n\} \subset W(k)$ and we see that $$ E_v = \bigcup\nolimits_{i = 1, \ldots, n} D j(w_i)^{-1} $$ in $G$ set theoretically. In particular we see that $G \setminus E_v$ is affine open (see above). Moreover, if $g \in G(k)$, then there exists a $v \in V$ such that $g \not \in E_v$. Namely, the set $W'$ of $w \in W$ such that $g \not \in Dj(w)^{-1}$ is nonempty open and it suffices to pick $v$ such that the fibre of $W' \to V$ over $v$ has $n$ elements. \medskip\noindent Consider the invertible sheaf $\mathcal{M} = \mathcal{O}_{G \times V}(E)$ on $G \times V$. By Varieties, Lemma \ref{varieties-lemma-rational-equivalence-for-Pic} the isomorphism class $\mathcal{L}$ of the restriction $\mathcal{M}_v = \mathcal{O}_G(E_v)$ is independent of $v \in V(k)$. On the other hand, for every $g \in G(k)$ we can find a $v$ such that $g \not \in E_v$ and such that $G \setminus E_v$ is affine. Thus the canonical section (Divisors, Definition \ref{divisors-definition-invertible-sheaf-effective-Cartier-divisor}) of $\mathcal{O}_G(E_v)$ corresponds to a section $s_v$ of $\mathcal{L}$ which does not vanish at $g$ and such that $G_{s_v}$ is affine. This means that $\mathcal{L}$ is ample by definition (Properties, Definition \ref{properties-definition-ample}).