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I am looking at Chapter IX (Compact Real Lie Groups), §4, Exercise 8 (translation). Given a complex subspace $\mathfrak p$ in the complexification $\mathfrak g_{\mathbf C}$ of some $\mathfrak g$, they ...

I am looking for ways to define $$\frac{1}{x}\in \mathbb{R}^n\quad \quad and\quad \quad x\cdot y\in \mathbb{R}^n ,$$ where $x,y\in \mathbb{R}^n$ such that $$\left\|\frac{1}{x}\right\|=\frac{1}{\|...