Let $H \subset G$ closed subgroup of an algebraic group. We want to prove the existence of the quotient $G/H$ which is a quasi-projective variety and homogeneous G-space.

We can find a vector $0 \ne y_0 = [v] \in \mathbb{P}^n$ such that $\forall h \in H: h\cdot y_0 = y_0$ and thus $H = \mathrm{Stab}_G(y_0) =G_{y_0}$ (stabilizer) and $G/H =$ the orbit of $y_0$ under $G$.

We define a group action G-morphism $\varphi: G \to Y$ by $g \mapsto g\cdot y_0$. We want to prove that the differential $(d \varphi)_e$ is separable and surjective.

The tangent space of an alebraic group is a derivative module $\mathrm{Der}_k(k[G],k)$ where $D \in \mathrm{Der}_k(k[G],k)$ is a k-linear derivation (i.e. $\forall x \in k: D(x)=Dx = 0$). We also define a derivation $\delta \in \mathrm{Der}_k(k[G],k[G])$ which returns a derivative function. We call $$\mathcal{L}(G) = \left( \{ \delta \in \mathrm{Der}_k(k[G],k[G]) | \delta \mbox{ is left invariant}, \delta \lambda_g(f(x)) = \lambda_g \delta f(x) . \} \right) $$ There is isomorphism between these spaces, $D \mapsto \delta_D$.

According to [Springer, "Linear Algebraic Group", 4.4.7, p.72] we deine $J$ to be the ideal in $k[G]$ of functions that vanish on $H$. We want to prove $$T_eH = ( \{ D \in T_eG | \delta_D I \subseteq I \} ) $$ (this is a set and a tagent space).

After that we want to show that $T_eH = \ker (d \varphi)_e$ and by a theorem it is equivalent that $\varphi$ is separable.

I tried but didn't managed to prove the last two assertions (the two equalities about $T_eH$), so I need some help with this.