  This paper discusses the construction of local bounded commuting projections
for discrete subcomplexes of the gradgrad complexes in two and three
dimensions, which play an important role in the finite element theory of
elasticity (2D) and general relativity (3D). The construction first extends the
local bounded commuting projections to the discrete de Rham complexes to other
discrete complexes. Moreover, the argument also provides a guidance in the
design of new discrete gradgrad complexes.
