Infinitesimal Zariski closures of positive representations

We classify the (semi-simple parts of the) Lie algebra of the Zariski closure of a discrete subgroup of a split simple real-algebraic Lie group, whose limit sets are minimal and such that the limit set in the space of full flags contains a positive triple of flags (as in Lusztig [23]). We then apply our result to obtain a new proof of Guichard’s classification [17] of Zariski closures of Hitchin representations into $\mathsf{PSL}_d (\mathbb{R})$.