  This paper presents the classification, over the fields of real and complex
numbers, of the minimal ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie algebras
and Lie superalgebras spanned by $4$ generators and with no empty graded
sector. The inequivalent graded Lie (super)algebras are obtained by solving the
constraints imposed by the respective graded Jacobi identities. A motivation
for this mathematical result is to sistematically investigate the properties of
dynamical systems invariant under graded (super)algebras. Recent works only
paid attention to the special case of the one-dimensional ${\mathbb
Z}_2\times{\mathbb Z}_2$-graded Poincar\'e superalgebra. As applications, we
are able to extend certain constructions originally introduced for this special
superalgebra to other listed ${\mathbb Z}_2\times{\mathbb Z}_2$-graded
(super)algebras. We mention, in particular, the notion of ${\mathbb
Z_2}\times{\mathbb Z}_2$-graded superspace and of invariant dynamical systems
(both classical worldline sigma models and a quantum Hamiltonian). As a further
byproduct we point out that, contrary to ${\mathbb Z}_2\times{\mathbb
Z}_2$-graded superalgebras, a theory invariant under a ${\mathbb
Z}_2\times{\mathbb Z}_2$-graded algebra implies the presence of ordinary bosons
and three different types of exotic bosons, with exotic bosons of different
types anticommuting among themselves.
