See Bourbaki, Groupes et algèbres de Lie, IV.2.7,  Corollary of Theorem 5.
 We assume that $K$ has at least four elements and that the Lie algebra is absolutely simple. We construct a $BN$-pair. Since the root system is irreducible, the Coxeter system is irreducible. If we denote by $G(K)^+$ the (normal) subgroup of $G(K)$ generated by the unipotent elements, the corollary says that  $G(K)^+/(G(K)^+\cap Z(K))$ is either a nonabelian simple group or reduces to one element. This proves that then the group $G(K)^+$  has no noncentral normal subgroups. When $K$ is algebraically closed, $K$ is infinite and  $G(K)^+=G(K)$.
Thus $G(K)$ has no noncentral normal subgroups, hence it has no nontrivial infinite normal subgroups. Thus $G$ has no nontrivial algebraic subgroups of positive dimension.