On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by
$$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [\bar w, -\bar z]\ ?$$

is this true for $\mathbb {CP}^n$?

The answer should be only one involution for even $n$, two of them for odd $n$.

The proof I have in mind uses the invariance of $z_i \mapsto M_{ij}\bar z_j$ under $U^{-1}MU^*$ together with the facts $MM^*=\pm 1$ and $MM^\dagger=1$ to get the conclusion, where $M^*$ is complex conjugate, while $M^\dagger$ is hermitian conjugate, i.e. complex conjugate and transpose, so it depends on the choice of Fubini-Study metric on $\mathbb{CP}^n$.

QUESTION: Is it possible to prove the statement without reference to the metric?