I assume it was paid for by the dotation royale, like the villa of Laurent was too (and all residences of the royals in Belgium I might add).
I however don't think that most Austrian cousins of Lorenz live in villa's of this size, the Habsburg family is not rich, it lost most after WWI and since there are so many of them the money and possessions that were still owned by them, had to be divided between the many children of emperor Karl and empress Zita. Most of these children have large families too, as do the grand children.
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