Definition :Let $V$ be a finite dimensional real vector space. A para-complex structure on $V$ is an endomorphism $K$ : $V \to V$ such that:

- $K$ is an involution, that is $K^2 = Id_V$ ;
- The eigenspaces $V := ker(Id_V \mp K)$ of $K$ with eigenvalues $1$ respectively have the same dimension. A vector space $V$ endowed with a para-complex structure $K$, denoted by $(V;K)$, will be called para-complex vector space. We know, we can identify $K$ with $K=\pmatrix{Id_n & \cr & -Id_n\cr} $ . Is there a basis of vectors like {$ x_1,x_2,...,x_n,y_1,y_2,...,y_n $}, such that $Kx_i=y_i$ and $Ky_i=x_i$. i.e. can we write is such basis $K=\pmatrix{ & Id_n \cr Id_n & \cr} $ ?