Embeddings of algebraic groups in Kac-Moody groups

Let $k_1,k_2$ be two fields of characteristic 0. Let $G_1$ be a split semisimple algebraic group over $k_1$, $G_2$ a split Kac–Moody group over $k_2$ and $\varphi \text{\hspace{0.17em}⁣}:G_1(k_1)\to G_2(k_2)$ an abstract embedding. We show that $\text{im}\varphi$ is a bounded subgroup whenever $k_1$ is an algebraic extension of the rational numbers, while there are embeddings with unbounded image if $k_1$ has infinite transcendence degree over the rational numbers.