Complexification and realification

One is asked to verify that an orthogonal operator $\mathcal{A}$ on a Euclidean vector space $V$, having no eigenvectors (this is possible only when $\operatorname {dim} V=2 m$), is the realification of a unitary operator $\mathcal{B}: U \rightarrow U$ on a complex vector space $U$ of dimension $m$ related to $V$. Note in this connection that the realification of a unitary space leads to a Euclidean space of twice the dimension.

Prove the formula from Proposition 3, choosing a basis $(\mathbf{e}_{1}, \ldots , \mathbf{e}_{n})$ of the complex space $U$ in which the matrix $A$ of the operator $\mathcal{A}: U \rightarrow U$ has upper triangular form with $\lambda_{1}, \ldots , \lambda_{n}$ on the diagonal.

Proposition 3: $\operatorname {det} A_{\mathbb {R}}=\left|\operatorname {det} A\right|^{2}$.

Let $U$ be a vector space over $\mathbb {C}$. What is the complexification $U^{\mathbb {C}}$ isomorphic to?

Let $(V,(* \mid *))$ be a Euclidean space, $V^{\mathbb {C}}$ its complexification, $\mathcal{A}$ a linear operator on $V^{\mathbb {C}}$, defined by the rule: $\mathcal{A}(\mathbf{u}+i \mathbf{v})=\mathbf{u}-i \mathbf{v}$ for all $\mathbf{u}, \mathbf{v} \in V$. Will $\mathcal{A}$ be a linear operator on the realification ${ }^{\mathbb {R}} V^{\mathbb {C}}$, and if so, will $\mathcal{A}$ be symmetric, orthogonal, idempotent?