Determine the exponential map of the direct product of two Lie groups.

Question

I know that the direct product of two Lie groups $G$ and $H$ is a Lie group.

Knowing the exponential map of $G$ and $H$, I would like to find an expression for the exponential map of $G \times H$.

I have that $\theta_X(t)=exp_G(tX)$ is the one parameter local subgroup associated to $X \in \mathfrak{g}$ and $\psi_Y(t)=exp_H(tY)$ is the one parameter local subgroup associated to $Y \in \mathfrak{h}$.

I have problems in finding the local subgroups associated to $(X,Y)$. Is it given by $t\rightarrow exp_G(tX) exp_H(tY)$?

Thanks for the help!

Answer 1

Let $G=G_2\times G_2$ with Lie algebras $\mathfrak{g}=\mathfrak{g}_1\times \mathfrak{g}_2$. Then we have, for $x_i\in \mathfrak{g}_i$ $$ \exp_G(x_1,x_2)=(\exp_{G_1}(x_1),\exp_{G_1}(x_2)) $$ as expected, see here.