Definitions
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be semisimple Lie algebras, and suppose that $(V,\rho)$ is a representation of $\mathfrak{g} = \mathfrak{g}_1 \oplus \mathfrak{g}_2$. Given a basis $\rho(x_1),\ldots, \rho(x_n)$ of $\rho(\mathfrak{g})$, let $\rho(y_1),\ldots, \rho(y_n)$ be the dual basis, in the sense that $\operatorname{tr}(\rho(x_i)\rho(y_j))$ is $1$ if $i=j$, and $0$ otherwise. Then we define the Casimir element of the representation to be $$ C_V = \sum_{i=1}^n\rho(x_i)\rho(y_i). $$ It is a standard result that this is a well-defined element of $\operatorname{End}_\mathfrak{g}(V)$.
Question
We may also view $V$ as a representation of $\mathfrak{g}_i$ for $i=1,2$, and we denote the corresponding Casimir elements by $C_V^i$. I would like to show that $$ C_V = C_V^1 + C_V^2. $$
My Attempt
I haven't made much meaningful progress, but I will list a couple of ideas here to show that I have at least thought about the problem. In the special case $\mathfrak{g}\subseteq \mathfrak{gl}(V)$, my guess is that we want to make a clever choice of dual pairs of bases of $\mathfrak{g}_1$ and $\mathfrak{g}_2$, such that their unions form a pair of dual bases for $\mathfrak{g}$. In order to do this, we might take a pair of bases for $\mathfrak{g}_1$, and then somehow extend them in a particular way to bases of $\mathfrak{g}$. I don't really have any ideas about the details of this, however. The general case seems more complicated, because we no longer have that $\rho(\mathfrak{g}) = \rho(\mathfrak{g}_1) \oplus \rho(\mathfrak{g}_2)$ in general.
Question: "Is the Casimir element of a semisimple Lie algebra is “additive”?"
Answer: The "universal Casimir elements" are certain elements in the center $Z(U(\mathfrak{g}))$ of the enveloping algebra of a semi simple Lie algebra. There is since $(V,\rho)$ is a representation, an action
$$\tilde{\rho}: U(\mathfrak{g})\rightarrow End_k(V)$$
And your "Casimir element" lives in $C_V\in End_k(V)$. Are you able to construct an element $\tilde{C}_V \in Z(U(\mathfrak{g}))$ mapping to your element? That is
$$\tilde{\rho}(\tilde{C}_V)=C_V.$$
The study of the center of the enveloping algebra is a much studied topic and the Harish-Chandra theorem may be of interest.
https://en.wikipedia.org/wiki/Universal_enveloping_algebra#Casimir_operators