Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map $f:\tilde F(P)\to F(P)$ which makes the diagram

\begin{array}{lll} \tilde F(P)\times ML(n,\mathbb{C}) & \rightarrow &\tilde F(P)\\ (f,r)\downarrow && \downarrow f\\ F(P)\times GL(n,\mathbb{C}) & \rightarrow & F(P) \end{array} commutes. The horizontal arrows are natural group actions and here $ML(n,\mathbb{C})$ is the metalinear group $\frac{\mathbb{C}\times SL(n,\mathbb{C})}{2\mathbb{Z}}$. We call this complex metalinear frame bundle a complex metalinear structure on $P$.

My question is when $M=\mathbb{C}P^n$, $M=\mathbb{C}P^{\infty}$ or $M=S^n$, what is the complex metalinear structure?

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map $f:\tilde F(P)\to F(P)$ and $r:ML(n,\mathbb{C})\to GL(n,\mathbb{C})$which makes the diagram

\begin{array}{lll} \tilde F(P)\times ML(n,\mathbb{C}) & \rightarrow &\tilde F(P)\\ (f,r)\downarrow && \downarrow f\\ F(P)\times GL(n,\mathbb{C}) & \rightarrow & F(P) \end{array} commutes. The horizontal arrows are natural group actions and here $ML(n,\mathbb{C})$ is the metalinear group $\frac{\mathbb{C}\times SL(n,\mathbb{C})}{2\mathbb{Z}}$. We call this complex metalinear frame bundle a complex metalinear structure on $P$.

My question is when $M=\mathbb{C}P^n$, $M=\mathbb{C}P^{\infty}$ or $M=S^n$, what is the complex metalinear structure?