So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly help me through).
As done previously, the group action defined by $h(g)p(z)=p(g^{-1}z)$ where $g\in\rm{SL}(3,\Bbb C)$, $p$ is from the vector space of polynomials of degree $\le2$ in three variables, and $z\in \Bbb C^3$.
I'm now introduced to a new function, $dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}$, where $X$ resides in the lie algebra of $\rm SL(3,\Bbb C)$ [ie $\mathfrak{sl}(3,\Bbb C)$]. The goal is: show that this is a lie algebra homomorphism $\mathfrak{sl}(3,\Bbb C)\to \mathfrak{gl}(6,\Bbb C)$.
Our basis in our space of polynomials is the standard one. Namely, the degree 2 terms in their various permutations.
I've been looking into the complexification of $\mathfrak{su}(3,\Bbb C)$ as a way of making sense of the polynomial when acted on by h, but I can't seem to get a handle on how to understand the derivative. I have a hunch this probably isn't even remotely the right course of action.
Any and all help is much appreciated. Feel free to assume I know the bare minimum.
Edit: While the induced homomorphism approach is awesome, a direct proof would help me get a better feel for how the matrix exponential is affecting the polynomial. Also, I'm too "machinery illiterate" to understand some of the more general formalisms at this point.