Embeddings of E(1,6) in E(5,10) and E(4,4)

We study the embeddings of the exceptional infinite-dimensional Lie superalgebra E(1,6) in the exceptional Lie superalgebras E(5,10) and E(4,4). These questions arose in the recent works on enhanced symmetries in some supersymmetric theories by N. Garner, S. Raghavendran, I. Saberi and B. Williams.