The Wikipedia entry on Lie Groups states (under the "Construction" section):
"The product of two Lie groups is a Lie group."
There is no further references or explanations about this statement.
What is the "product" that this refers to?
For example, given 2 sub-groups of $GL(3,\Bbb R)$, say $G_1\equiv SO(3)$ and $G2\equiv UT(3)$, the invertible upper triangular matrices, they both have a compatible inner-product.
Does the above statement mean that $G_1\times G_2$ (or is it $G_1\cdot G_2$) is a Lie sub-group itself?
This would be easy to show if $G_1$ and $G_2$ were commutable, but they aren't.
This is the direct product: $G \times H$, which is the set of $\{(g,h) \mid g \in G, h \in H\}$ with $(g_1,h_1) \cdot_{G \times H} (g_2,h_2) = (g_1 \cdot_G g_2, h_1 \cdot_H h_2)$, i.e., multiplication is elementwise. The identity is the ordered pair of the identity from each of $G$ and $H$. Inverses are elementwise.