This question is about the relationship (rather, whether there is or ought to be a relationship) between torsion for the cohomology of certain Lie algebras over the integers, and torsion for associated Lie groups.

Let $\mathfrak{g}$ be a complex simple Lie algebra. Then we can choose a basis of Chevalley generators for $\mathfrak{g}$. The structure constants describing the Lie brackets among the Chevalley generators are all integers, so the Chevalley basis spans a Lie algebra over the integers, which I shall denote by $\mathfrak{g}_\mathbb{Z}$. We can compute the (Chevalley-Eilenberg) Lie algebra cohomology of $\mathfrak{g}_\mathbb{Z}$ with coefficients in $\mathbb{Z}$ using the usual Koszul complex, and thus get for each $1 \leq n \leq \dim \mathfrak{g}$ a finitely-generated abelian group $H^n(\mathfrak{g}_\mathbb{Z},\mathbb{Z})$.

On the other hand, (as I understand it) the Lie algebra cohomology of $\mathfrak{g}$ with coefficients in $\mathbb{C}$, denoted $H^*(\mathfrak{g},\mathbb{C})$, is the same as the cohomology of some associated (compact?) connected Lie group $G$. This philosophy is described in the section "Motivation" of the Wikipedia article on Lie algebra cohomology. (I'm not an expert on this matter, so I invite someone more knowledgable to correct me if I've got something incorrect. I think that the associated Lie group $G$ will have $\mathfrak{g}$ as its Lie algebra, or perhaps $\mathfrak{g}$ will be the complexification of the Lie algebra of $G$.) We can also compute $H^n(G,\mathbb{Z})$, the cohomology of $G$ with coefficients in $\mathbb{Z}$.

Is there, or ought there to be, a connection between the torsion of $H^n(G,\mathbb{Z})$ and the torsion of $H^n(\mathfrak{g}_\mathbb{Z},\mathbb{Z})$? Should the torsion primes of the two abelian groups be the same? Should the torsion primes of one be a subset of the torsion primes for the other?

To the best of my knowledge, the torsion primes of the compact connected simple Lie groups were worked out in the 1950s and/or 1960s, especially by Armand Borel, but at present no such list seems readily available for the Lie algebra cohomology side of the picture.