Is the product of embedded submanifolds still an embedded submanifold? My terminology complies with Lee's ISM. Let $G$ be a Lie group and suppose $H$ is a subgroup of $G$ that is also an embedded submanifold. I would like to show that $H$ is a Lie subgroup. To begin with, I wonder if $H\times H$ is an embedded submanifold of $G\times G$. In order to answer this question, one needs to check if the inclusion map $\iota_2:H\times H\hookrightarrow G\times G$ is a smooth embedding. And that's where I got stuck. The inclusion map $\iota_1:H\hookrightarrow G$ is a smooth embedding by hypothesis, but I have no idea how to relate this fact to $\iota_2$. Thank you.
 A: We show the following result.

Let $f_1 \colon N_1 \to M_1$ and $f_2 \colon N_2 \to M_2$ be two smooth embeddings. Then the map
\begin{align}
f\colon N_1\times N_2 & \longrightarrow M_1\times M_2 \\
(x,y) & \longmapsto (f_1(x),f_2(y))
\end{align}
is an embedding.

First note that we have the identifications
\begin{align}
T_{(x,y)}\left(N_1\times N_2\right) &\simeq T_xN_1 \times T_yN_2, & T_{f(x,y)}\left(M_1\times M_2\right) \simeq T_{f_1(x)}M_1 \times T_{f_2(x)}M_2.
\end{align}
Therefore, $\mathrm{d}f$ can be written at a point $(x,y)$ as
\begin{align}
\mathrm{d}f(x,y) \colon T_x N_1 \times T_yN_2 & \longrightarrow T_{f_1(x)}M_1 \times T_{f_2(y)}M_2 \\
(v,w) & \longmapsto \left(\mathrm{d}f_1(x)v, \mathrm{d}f_2(y)w\right) 
\end{align}
As $f_1$ and $f_2$ are supposed to be embeddings, they are in particular immersions, so that $\mathrm{d}f_1(x)$ and $\mathrm{d}f_2(y)$ are injective linear maps. It follows that so is $\mathrm{d}f(x,y)$. Therefore, $f$ is an immersion.
In order to conclude, we have to show that $f$ is a homeomorphism onto its image, that is, it has a continuous inverse. Let $g_1 \colon f_1(N_1) \to N_1$ be the inverse homeomorphism of $f_1\colon N_1 \to f_1(N_1)$, and $g_2\colon f_2(N_2) \to N_2$ whose of $f_2\colon N_2 \to f_2(N_2)$. They are continuous by hypothesis. Define $g(a,b) = (g_1(a),g_2(b))$ for $(a,b) \in f_1(N_1)\times f_2(N_2) \subset M_1\times M_2$. Check that $g\circ f = \mathrm{Id}_{N_1\times N_2}$ and $f\circ g = \mathrm{Id}_{f_1(N_1)\times f_2(N_2)}$. It follows that $f$ has a continuous inverse, and $f$ is a homeomorphism onto its image.
Finally, $f$ is an immersion which is a homeomorphism onto its image: it is an embedding.

Getting back to the original question. Apply the above result to $N_1=N_2 = H$, $M_1=M_2=G$, $f_1=f_2=\iota_1\colon H \to G$: it follows that $H\times H$ is an embedded submanifold of $G\times G$.
