We study the branching of representations of a $p$-elementary quadratic form by a genus of positive definite locally $p$-two-dimensional forms. A primitive representation of a $p$-elementary form is decomposed into a direct sum of minimal indecomposable representations; the latter representations are found in an explicit form. For the case of branching, we find local multipliers of the weight of representations of a form by a genus. As an application, we calculate the number of embeddings into the classical root lattices. The method of orthogonal complement is applied in constructing new genera of quadratic forms.