Let $V$ be $n$-dimensional $\mathbb{R}$-vector space with inner product $\langle \,\,,\,\,\rangle_1$. Let $W=V\oplus V$. Then $W$ can be made into a $\mathbb{C}$-vector space by defining addition as usual but scalar multiplication by $(a+ib)(w_1,w_2)=(aw_1-bw_2, aw_2+bw_1)$ where $(w_1,w_2)\in V\oplus V$ (here $w_1$ is in first component $V$, $w_2$ is in second component $V$.)
Question 1. Is it true that $\langle \,\,,\,\,\rangle_1$ can be extended to an inner product $\langle \,\,,\,\,\rangle_2\colon W\times W\rightarrow \mathbb{R}$? Is the extension unique?
