The orthogonal group $\mathcal O_n (\mathbb K)$ is isomorphic to a subgroup of $\mathcal O_{n+1} (\mathbb K)$

Question

I tried to do the following exercise:

Prove that $\mathcal O_n (\mathbb K)$ is isomorphic to a subgroup of $\mathcal O_{n+1} (\mathbb K)$

The definition of $\mathcal O_n (\mathbb K)$ is

$$\mathcal O_n (\mathbb K) =\{A \in M_n(\mathbb K) \mid \langle Ax , Ay\rangle = \langle x,y\rangle \}$$

My answer is: Note that $$ \left\{ \begin{pmatrix} 1 & 0 \\ 0 & A \end{pmatrix} \mid A \in \mathcal O_{n} \right\}$$

is a subgroup with the desired property.

Is it really so easy or am I missing something?

Answer 1

Yes, it is really so easy. ${}$