Let \( G \) be a simple algebraic group and \( \mathcal O \subset \mathfrak g = Lie(G) \) a nilpotent orbit. If \( H \) is a reductive subgroup of \( G \), then \( \mathfrak g = \mathfrak h \oplus \mathfrak m \), where \( \mathfrak m = \mathfrak h^\perp \). We consider the natural projections \( \varphi : \overline{\mathcal O} \to \mathfrak h \) and \( \psi : \overline{\mathcal O} \to \mathfrak m \) and two related properties of \( (H, \mathcal O) \): \( (P_1) \): \( \mathcal O \cap \mathfrak m = \{0\} \); \( (P_2) \): \( H \) has a dense orbit in \( \mathcal O \). We prove that \( (P_1) \Rightarrow (P_2) \) for all \( \mathcal O \) and the converse holds for \( \mathcal O_{min} \), the minimal nilpotent orbit. If \( (P_1) \) holds, then \( \varphi \) is finite and \( [\varphi(e), \psi(e)] = 0 \) for all \( e \in \mathcal O \). Here \( \overline{\varphi(\mathcal O)} \) is the closure of a nilpotent \( H \)-orbit \( \mathcal O' \) and the orbits \( \mathcal O \) and \( O' \) are “shared” in the sense of Brylinski–Kostant (1994). We provide a classification of all pairs \( (H, \mathcal O) \) with property \( (P_1) \) and discuss relations between \( O \) and \( O' \). In particular, we point out an omission in the list of “shared” orbits. This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.