Reference: Peter Petersen, Riemannian Geometry, 3rd edition, Example 1.1.5
Hopf fibration $F: S^3(1) \to S^2 (1/2)$ is defined by $$F(z,w) = \left(\frac{1}{2} (|w|^2 - |z|^2), z\overline{w}\right)$$ if we think of $S^3(1) \subset C^2$ and $S^2(1/2) \subset R \oplus C$. I want to show that $F$ is a Riemannian submersion.
Definition: a Riemannian submersion $F:(M,g_M) \to(N,g_N)$ is a submersion $F:M \to N$ such that for each $p\in M$, $DF:\ker(DF)^\perp \to T_{f(p)}N$ is a linear isometry. That is, if $v,w \in T_pM$ are perpendicular to the kernel of $DF: T_pM \to T_{F(p)N}$, then $$g_M(v,w) = g_N(DF(v),DF(w)).$$
To show that $F$ is a Riemannian submersion, I tried to find $\ker (DF)$. Since I don't know the method to find $DF$ when $F$ is not a real function, I set $F$ as a real-valued function as below. $$F(a,b,c,d) = \left(\frac{1}{2} ((a^2 - b^2) - (c^2 -d^2)), ac+bd, bc-ad\right).$$ Then I computed $$DF =\begin{pmatrix} a & -b & -c & d \\ c & d & a & b \\ -d & c& b& -a\end{pmatrix}$$ To find a kernel, I used elementary row operations to an augmented matrix $(DF|0)$. This calculation is very messy, so I am not sure whether this method is right or not.
p.s. I know there are similar questions( The hopf fibration is a submersion. Hopf fibration is a submersion ), but I didn't understand their answers. Moreover, What I want to do is to find $\ker(DF)$.
