Is this a typo in the retraction formula?

Question

In the book, optimization on matrix manifolds by Absil, Chapter 3 pg 59, it is given that retraction on the orthogonal manifold $O_n$, using the Cayley's transform, is given by $$ R_{X}(X \Omega)=X\left(I-\frac{1}{2} \Omega\right)^{-1}\left(I+\frac{1}{2} \Omega\right) $$ I am trying to derive to this from first principles but stuck mid-way.

My attempt:

Let $Q \in O_n$ and $\Omega$ be a skew symmetric matrix. By Cayley's transform, for each $\Omega$ we can find a $Q$ such that $$Q = \left( I - \Omega \right) \left( I + \Omega \right)^{-1}$$ Let $\phi$ be the mapping from $[Q^{-1},I - \Omega] \rightarrow I+\Omega$. Therefore, we can write $\pi_{1} \circ \phi^{-1}$ as matrix that takes in $I+\Omega$ and outputs $Q^{-1}$.

\begin{align} R_{X}(X+X\Omega) &:=\pi_{1}\left(\phi^{-1}(X+X\Omega)\right) \\ & = X \pi_{1}\left(\phi^{-1}(I+\Omega)\right) \\ & = X Q^{-1} \\ & = X \left( I + \Omega \right) \left( I - \Omega \right)^{-1} \\ & = X \left( I - \Omega \right)^{-1} \left( I + \Omega \right) \qquad \quad \text{as the matrices commute}\\ \end{align}

I am not sure how to get the the $\frac{1}{2}$ as mentioned in the textbook. Can you please help. Thank you.

Update: One of the answers says that this is a typo in the book. Can others please confirm?

Below is a picture of the relevant text and symbols.

Answer 1

Hint: Let $\Omega$ be skew-symmetric of order $n$. Then $\frac{1}{2} \Omega$ is also skew-symmetric. Hence, by Cayley's transform there exists a $Q \in O_n$ such that $$Q = (I - \tfrac{1}{2} \Omega) (I + \tfrac{1}{2} \Omega)^{-1}$$.

Answer 2

An errata for Optimization Algorithms on Matrix Manifolds can be found at https://sites.uclouvain.be/absil/amsbook/errata.html. So far, the Cayley retraction formula on top of page 59 has been considered to be correct.

I doubt that Proposition 4.1.2 can be used here. The first step would be to clarify what $\mathcal{N}$ is, and to make sure that $\phi$ has the required domain and codomain.

Hint: A way to prove that $$ R_{X}(X \Omega)=X\left(I-\frac{1}{2} \Omega\right)^{-1}\left(I+\frac{1}{2} \Omega\right) $$ is a retraction on the orthogonal group is to confirm that it satisfies the properties given in Definition 4.1.1. The hardest part is arguably (ii) therein. Using the product rule, the chain rule and (A.3) should yield the sought result.