— Cartan matrix for an exotic type of Lie algebra —

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra?

For example, consider this Lie algebra:

$$
[X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = – f_{ik}{}^j Y^k \qquad\qquad [Y^i,Y^j] = 0
$$
Here, 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 Abelian extension of $X_1,X_2,X_3$. (Some people would use the words semi-direct product for the groups $g(X) \ltimes g(Y)^*$.)

(2) What is the Cartan matrix for this Lie algebra above? (for this whole non-semisimple Lie algebra $g(X) \ltimes g(Y)^*$.)

Thank you for any comments and concerns! Please reply whatever thoughts you have.

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