The question is a modified one inspired by this post:
What is the Cartan matrix for this Lie algebra below? (for this semisimple Lie algebra $g(X) \oplus h(Y)$,)
$$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [Y_a,Y_b] = F_{ab}{}^c Y_c \qquad\qquad [X_i,Y_a] = \mathcal{F}_{ia}{}^k X_k $$ Here, $f_{ij}{}^k, F_{ab}{}^c, \mathcal{F}_{ia}{}^k$ are three different structure constants. And $i,j,k \in \{1,2,3\}$, $a,b,c \in \{1,2,3\}$; there are 3 generators $X_1,X_2,X_3$ and 3 generators $Y_1,Y_2,Y_3$.
If there is a generic form of Cartan matrix for this algera will be even better.
If not, we may, for example, consider 3 generators $X_1,X_2,X_3$ generate a compact semi-simple SU(2) Lie algebra with $f_{ij}{}^k$ given by $f_{12}{}^3=1$ and $f_{23}{}^1=-1$ as $i,j,k$ are cyclic. And another 3 generators $Y^1,Y^2,Y^3$ are extension of $X_1,X_2,X_3$. We may also consider also $F_{ab}{}^c$ and $\mathcal{F}_{ia}{}^k $ are structure constants of SU(2) Lie algebra.
I suppose this modified Lie algebra is still semisimple Lie algebra, because there is no nontrivial maximal solvable Ideal.
Thank you for any comments and concerns. Please provide whatever thoughts!
This depends on the value of ${\mathcal F}$. You will either get direct sum of two su(2)'s or a single su(2).
Here are some details: I will assume that $f$ and $F$ are structure constants of a simple Lie algebra, say $su(2)$ (using $sl(2)$ will make very little difference). Observe also we have a homomormpism $h: su(2)\to {\mathfrak g}$ with target the Lie algebra ${\mathfrak g}$ you defined, sending generators of $su(2)$ to the vectors $X_i$. Observe that there is no a priori reason for this homomorphism to be injective; by simplicity of $su(2)$, this homomorphism is either zero or injective. The image $h(su(2))$ is an ideal in ${\mathfrak g}$; dividing by this ideal we obtain Lie algebra with structure constants $F$; by my assumption, this is again $su(2)$. Thus, we obtain a split exact sequence $$ su(2) \to {\mathfrak g} \to su(2)\to 0. $$ This sequence splits (either by assumption as in your question or because $su(2)$ is semisimple); hence, we obtain $$ {\mathfrak g}\cong h(su(2))\oplus su(2) $$ Hence, ${\mathfrak g}$ is either isomorphic to $su(2)$ or to $su(2)\oplus su(2)$. Which case happens depends on ${\mathcal F}$. If indeed ${\mathcal F}$ are the structure constants of $su(2)$ then we also obtain a surjection $$ r: {\mathfrak g}\to su(2) $$ such that $r(X_i)=r(Y_i)$, in which case $h$ cannot be zero. Hence, in this case, $$ {\mathfrak g}\cong su(2)\oplus su(2). $$ Its Cartain matrix is, of course, just the direct sum of Cartan matrices of two $su(2)$'s.