Given the functions $f_i : \mathbb{R}^4 \to \mathbb{R}, i = 1, 2, 3$ with
$f_1(x) = x_1x_3 - x_2^2$
$f_2(x) = x_2x_4 - x_3^2$
$f_3(x) = x_1x_4 - x_2x_3$
show that $M:= \{ x \in \mathbb{R}^4\setminus\{0\}: f_1(x) = f_2(x) = f_3(x) = 0 \}$ is a 2-dimensional submanifold of $\mathbb{R}^4$
So I have to show that for all points $m \in M$ where is an open environment $U$ and two continuously differentiable function $g_1, g_2: U \to \mathbb{R}$ such that $M \cap U = \{x \in U : g_1(x) = g_2(x) = 0 \}$ and $ \operatorname{rank} \frac{\partial(g_1, g_2)}{\partial(x_1, x_2, x_3, x_4)} = 2 \; \forall x \in M \cap U$.
Obvious candidates for $g_1$ and $g_2$ are the $f_i$ ($i = 1,2,3$) which would have to be chosen depending on the $m \in M$. So firstly I'd have to find an environment for each $m$ where setting two of the three $f_i$ to zero implies that the third one is zero and then show the rank condition.
Any ideas?