This paper continues the study of associative and Lie deep matrix algebras, DM(X, K) and gld(X, K), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras, BDM(X, K) and bld(X, K), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal lattices. In particular, bld(X, K) is shown to be semisimple. The Lie algebra bld(X, K) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct product of sl(n)'s. We classify all associative bilinear forms on sl(2)d (a natural depth analogue of sl(2)) and bld.