It is well known that $SU(3)$ is the unique, non-trivial, principal $SU(2)$- bundle over $S^5$. To my knowledge the way this is proven is by using the following fact: 



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> If $G$ is a Lie group and $H$ a closed subgroup, then $G \to G/H$ is a (locally trivial) principal $H$-bundle. 
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In our case, $SU(3)$ acts transitively on $S^5 \subseteq \mathbb C^3$, the stabilizer of any point is $SU(2)$, and we use orbit-stabilizer to conclude the desired property. The remaining part is a simple computation of  

$$[S^5,BSU(2)] = \pi\_5(BSU(2)) = \pi\_4(\Omega BSU(2)) = \pi\_4(SU(2)) = \pi\_4(S^3) = \mathbb Z/2\mathbb Z.$$ 
And $SU(3)$ is not the trivial one since $\pi\_4(SU(3)) \neq \pi\_4(SU(2))$. 


Unfortunately, I am interested in writing down an explicit trivialization for which this method does not seem to provide any insight. In particular, I would like to trivialize $SU(2) \hookrightarrow SU(3) \rightarrow S^5$ over two open sets given by removing distinct points. More precisely, if $N\_\pm=(\pm1,0,0) \subseteq \mathbb C^3$, let's trivialize over $U\_\pm = S^5 \setminus\{N\_\pm\}$.


**Attempt One:** The naive procedure to begin with is to view $SU(3)$ as the collection of orthonormal bases $\{(e\_1,e\_2,e\_3)\}$ for $\mathbb C^3$. Let us then take the projection $SU(3) \to S^5$ to be $p(e\_1,e\_2,e\_3) = e\_1$. The problem is then to map $(e\_2,e\_3)$ to some element in $SU(2)$ given that $e\_1 \neq N\_\pm$. I cannot see what that map should be though.


**Attempt Two** My second attempt is to emulate the solution to [this overflow post](https://mathoverflow.net/questions/139987/a-concrete-realization-of-the-nontrivial-2-sphere-bundle-over-the-5-sphere), which would have me define the set $X$ as the set of pairs $(v,A)$ where $v \in S^5 \subseteq \mathbb C^3$ and $A \in (v^\perp)^1 \cong S^3 \cong SU(2)$, the unit sphere in $v^\perp$. This is certainly a principal $SU(2)$-bundle, and while I have yet to show it is non-trivial I believe it is as such. As the non-trivial $SU(2)$-bundle is unique up to isomorphism, there is an $SU(2)$-equivariant diffeomorphism $X \to SU(3)$, but I am not certain as how to find such an isomorphism. Any suggestions are appreciated. 



Local trivializations of the non-trivial $SU(2)$-bundle over $S^5$
The problem seems to be one of systematically extending $z=(z\_1,z\_2,z\_3)\in S^5$ to an orthonormal basis of $\mathbb{C}^3$. To this end, let $e\_1,e\_2,e\_3$ be the standard basis vectors for $\mathbb{C}^3$. For $j\in\{1,2,3\}$, let $$U\_j:=\{z\in S^5:\{z,e\_1,e\_2,e\_3\}\setminus\{e\_j\}\text{ is a basis of }\mathbb{C}^3\}.$$ This is an open cover of $S^5$. 


Taking determinants, we find that $U\_j=\{z\in S^5:z\_j\neq 0\}$. Perhaps we can trivialize your bundle over the three open sets $U\_j$. For $z\in U\_j$, I would suggest using Gram-Schmidt on the basis $\{z,e\_1,e\_2,e\_3\}\setminus\{e\_j\}$ to obtain $\varphi\_j(z):=[z\text{ }f\_{1j}(z)\text{ }f\_{2j}(z)]\in SU(3)$, where $f\_{1j},f\_{2j}:U\_j\rightarrow\mathbb{C}^3$. The sections $\varphi\_j$ should be computable and they should allow you to trivialize the bundle over the open sets $U\_j$. 


I realize that I have not given local trivializations over the complements of the north and south poles. Still, it is some local trivialization.


