In my question I just want to find an example of f_1 and f_3 and a real four dimentional subspace H such that $f=(e^{f_1},−e^{f_1},e^{f_3})$ avoid H. we can choose for example $f_3=2f_1$ or $f_3=0$ and $f_1=2z$ because we just want an example.

no sorry for the mistake it should be $Span_\mathbb{R}(H^\perp,H_j,H_k)=\mathbb{R}^6$