The only hopf fibrations that exist are the first three listed here.
$\require{AMScd}$ \begin{CD} S^3 @>S^1>> S^2 \end{CD}
$\require{AMScd}$ \begin{CD} S^7 @>S^3>> S^4 \end{CD}
$\require{AMScd}$ \begin{CD} S^{15} @>S^7>> S^8 \end{CD}
For a general case of SU(n) the fibrations can be written as such : $\require{AMScd}$ \begin{CD} S^{2n^2-1} @>S^{2n-1}>> S^{2n[n-1]} \end{CD}
$\require{AMScd}$ \begin{CD} S^{2n^3-1} @>S^{2n^2-1}>> S^{2n^2[n-1]} \end{CD} For the case of n = 3,These are fibrations that should theoretically exist. Is there a map that can be written for this case (somethink akin to complex numbers,quaternions and octonions) ? $\require{AMScd}$ \begin{CD} S^{17} @>S^5>> S^{12} \end{CD}
$\require{AMScd}$ \begin{CD} S^{53} @>S^{17}>> S^{36} \end{CD}