Deformation subspaces of $p$-divisible groups as formal Lie groups associated to $p$-divisible groups

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$ which is not isoclinic. Let $\mathcal {D}$ (resp. $\mathcal {D}_{k}$) be the formal deformation space of $D$ over $\text {Spf}(W(k))$ (resp. over $\text {Spf}(k)$). We use axioms to construct formal subschemes $\mathcal {G}_{k}$ of $\mathcal {D}_{k}$ that: (i) have canonical structures of formal Lie groups over $\text {Spf}(k)$ associated to $p$-divisible groups over $k$, and (ii) give birth, via all geometric points $\text {Spf}(K)\to \mathcal {G}_{k}$, to $p$-divisible groups over $K$ that are isomorphic to $D_{K}$. We also identify when there exist formal subschemes $\mathcal {G}$ of $\mathcal {D}$ which lift $\mathcal {G}_{k}$ and which have natural structures of formal Lie groups over $\text {Spf}(W(k))$ associated to $p$-divisible groups over $W(k)$. Applications to Traverso (ultimate) stratifications are included as well.