Identifying a specific $\operatorname{Sp}(4,\mathbb{C})$-representation

Question

Let $V$ be some given vector space and $\operatorname{Sp}(4,\mathbb{C})\rightarrow \operatorname{Gl}(V)$ be a finite dimensional representation. Let $H\subset \operatorname{Sp}(4,\mathbb{C})$ be the subgroup of all diagonal matrices. I need to show that the representation in question is the irreducible representation corresponding to the second fundamental weight of $\operatorname{Sp}(4,\mathbb{C})$. I think know how to do this for Lie algebra representations, but I am not sure about how this translate to group representations.

The roots of the symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{C})$ are $\pm L_i \pm L_j$ for $i,j =1,2$ with $L_i\colon \mathfrak{h}\rightarrow \mathbb{C}$, $L_i(a_{ii})= a_{ii}$ for any diagonal symplectic matrix $(a_{ii})\in\mathfrak{h}$. We can choose a Weyl chamber such that the positive roots are $L_1+L_2$, $2L_1$, $2L_2$ and $L_1-L_2$. The primitive positive roots are $L_1-L_2$ and $2L_2$. Denote by $\mathfrak{g}_\alpha$ the root space of $\mathfrak{sp}(4,\mathbb{C})$ corresponding to a root $\alpha$.

1. First, am I correct in that the second fundamental weight is the map $\lambda_2\colon H\rightarrow \mathbb{C}, \lambda_2(a_{ii})=a_{11}a_{22}$ where $(a_ii)$ is a diagonal matrix in $H$?
2. To show that the given representation above contains the irreducible representation corresponding to $\lambda_2$, do I need to find a vector $v\in V$ such that $h\cdot v = \lambda_2(h)$ for all $h\in H$ and $X\cdot v = 0$ for all $X$ in $\mathfrak{g}_\alpha$ for positive roots $\alpha$ of $\mathfrak{sp}(4,\mathbb{C})$?
3. If I have found such a vector, to show that the representation is irreducible, do I need to show that I can generate all of $V$ by successive applications of elements of $\mathfrak{g}_\beta$ where $\beta$ are primitive negative roots?
4. Are the primitive negative roots $-2L_2$ and $L_2-L_1$?
5. I know that for example $U=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} $ is contained in $\mathfrak{g}_{2L_1}$. How do I find all other elements of this root space so that I can show that indeed $X\cdot v =0$ for all $X\in\mathfrak{g}_{2L_1}$?

If one step in the above is incorrect, how do I show that the given finite dimensional representation is the irreducible one corresponding to the second fundamental weight?