Proposition 3

Let $G_1$ and $G_2$ be two finite dimensional Lie groups over $k$. An element $(g_1, g_2)$ of $G_1\times G_2$ is regular if and only if $g_1$ and $g_2$ are regular elements of $G_1$ and $G_2$, respectively.

The condition is necessary by Prop. 2. We show that it is sufficient. For all $g= (g_1, g_2)\in G_1\times G_2,r_{Ad}(g) =r_{Ad}(g_1) +r_{Ad}(g_2)$. In view of Lemma 1 (ii), it follows that $r^0_{Ad}(g) =r^0_{Ad}(g_1) +r_{Ad}^0(g_2)$. If $g_1$ and $g_2$ are regular, $r^0_{Ad}(g_1) =$ $r_{Ad}(g_1)$ and $r^0_{Ad}(g_2) =r_{Ad}(g_2)$, so $r^0_{Ad}(g) =r_{Ad}(g)$, which means that $g$ is regular.