Solution
= Solution
Ambient matrix multiplication
$$
M_{2n}(\mathbb R)\times M_{2n}(\mathbb R)\longrightarrow M_{2n}(\mathbb R),
\qquad(A,B)\longmapsto AB,
$$
is bilinear and therefore smooth. Its restriction to the embedded submanifold $G\times G$ is smooth, and part a shows that its image lies in $G$. By the defining smooth structure on an <embedded submanifold>, this restriction is a smooth map $G\times G\to G$.