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

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.
