Real projective space has fixed-point property iff it has even dimension

Statement
Suppose $$n$$ is a natural number. Consider the fact about::real projective space $$\mathbb{P}^n(\R)$$ (also denoted $$R\mathbb{P}^n$$) of dimension $$n$$. Note that this can be interpreted as the set of nonzero vectors, up to scalar multiplication equivalence, in $$\R^{n+1}$$, and its elements can be written in the form $$[a_1:a_2:\dots:a_n:a_{n+1}]$$ with all $$a_i \in \R$$ and not all of them simultaneously zero, where:

$$[a_1:a_2:\dots:a_n:a_{n+1}] = [b_1:b_2:\dots:b_n:b_{n+1}] \iff a_ib_j = a_jb_i \ \forall \ i,j \in \{ 1,2,3,\dots,n+1 \}$$

The claim is the following:


 * If $$n$$ is odd (i.e., $$n = 1,3,5,\dots$$), then $$\mathbb{P}^n(\R)$$ does not have the fact about::fixed-point property, i.e., we can find a continuous map $$f: \mathbb{P}^n(\R) \to \mathbb{P}^n(\R)$$ such that $$f$$ does not have any fixed point. In fact, we can choose the continuous map to be a self-homeomorphism and even to be an algebraic automorphism.
 * If $$n$$ is even (i.e., $$n = 2,4,6,\dots$$), then $$\mathbb{P}^n(\R)$$ has the fixed-point property, i.e., for any continuous map from $$\mathbb{P}^n(\R)$$ to itself, there is a fixed point.

Similar facts that distinguish real projective spaces of even and odd dimension

 * Odd-dimensional real projective spaces are orientable, even-dimensional real projective spaces are non-orientable. See homology of real projective space and cohomology of real projective space.
 * Odd-dimensional real projective spaces also possess an orientation-reversing homeomorphism. The question doesn't even arise for even-dimensional real projective space.

Similar facts about complex and quaternionic projective space

 * Complex projective space has fixed-point property iff it has even complex dimension
 * Complex projective space has orientation-reversing self-homeomorphism iff it has odd complex dimension

Facts used

 * 1) uses::Rationally acyclic compact polyhedron has fixed-point property, which in turn follows from the uses::Lefschetz fixed-point theorem.
 * 2) uses::Homology of real projective space, whereby it's clear that the even-dimensional real projective spaces are rationally acyclic.
 * 3) Real projective spaces are compact polyhedra.

Even dimension
The proof in even dimensions follows directly by combining Facts (1), (2), and (3).

Odd dimension
In odd dimension, we can explicitly construct an algebraic self-map $$f:\mathbb{P}^n(\R) \to \mathbb{P}^n(\R)$$ that works:

$$[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-a_2:a_1:-a_4:a_3:\dots:-a_{n+1}:a_n]$$

Note that we are using that $$n$$, or equivalently, that $$n+1$$ is even, to make sense of the above.

We verify: