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 $\scrD$ (resp. $\scrD_k$) be the formal deformation space of $D$ over $\Spf(W(k))$ (resp. over $\Spf(k)$). We use axioms to construct formal subschemes $\scrG_k$ of $\scrD_k$ that: (i) have canonical structures of formal Lie groups over $\Spf(k)$ associated to $p$-divisible groups over $k$, and (ii) give birth, via all geometric points $\Spf(K)\to\scrG_k$, to $p$-divisible groups over $K$ that are isomorphic to $D_K$. We also identify when there exist formal subschemes $\scrG$ of $\scrD$ which lift $\scrG_k$ and which have natural structures of formal Lie groups over $\Spf(W(k))$ associated to $p$-divisible groups over $W(k)$. Applications to Traverso (ultimate) stratifications are included as well.