compactness or not of a Lie group

Question

1. Is the Lie group generated by this Lie algebra compact or not?

$$ [X_i,X_j]=0, [H_i,H_j]=f^{ijk} X_k, [X_i,H_j]=0 $$

$f^{123}>0$, and $i,j,k \in \{ 1,2,3\}$. There are 6 generators in total for the Lie algebra.

2. What is the easy way to check the compactness of this Lie group? Is the Lie group unique or not? If there is a unique simply connected Lie group associated to a (finite-dimensional) Lie algebra, then whether the compactness of this simply connected Lie group unique determined?

Any constructive comment or answer will be helpful.

Answer 1

I suppose the invariant bilinear matrix replacing the degenerate Killing form for this Lie algebra case is:

$$ \begin{pmatrix} 0 & 0& 0 & 1 & 0& 0\\ 0 & 0& 0 & 0 & 1& 0\\ 0 & 0& 0 & 0 & 0& 1\\ 1 & 0& 0 & 0 & 0& 0\\ 0 & 1& 0 & 0 & 0& 0\\ 0 & 0& 1 & 0 & 0& 0 \end{pmatrix} $$ where the 1, 2, 3, 4, 5,6 components correspond to $X1,X2,X3,H1,H2,H3$ respectively.

So the invariant bilinear matrix is not positive definite, its eigenvalues are 1,1,1,-1,-1,-1,; so the corresponding Lie group will be non-compact.