  We start with a realisation of a Lie algebra with the basis operators
$L=\langle Q_m\rangle$, $Q_m=\zeta_{mj}(x_i)\partial_{x_j}$, where $x_i$ are
some variables that may be regarded as dependent or independent in construction
of some equations or differential invariants. We take additional variables
$R_k$, and study the linearly extended action operators $ {\hat
Q_m}={Q_m}+\lambda_{mjk}R_j\partial_{R_k}$ that form the same Lie algebra with
the same structural constants. For a fixed realisation of any Lie algebra $L$
we can classify all inequivalent extended action realisations for a finite
number of additional variables. Such realisations allow to construct new
invariant equations for the same realisation of the algebra, but involving
additional variables, and to classify exhaustively relative differential
invariants and invariant equations for the respective realisations of Lie
algebras. They can be also applied to other problems in the symmetry analysis
of differential equations. Here we classify extensions of realisations for
inequivalent low-dimensional Lie algebras. The problem of classification of the
relative differential invariants has an interesting story attached to it, and
the background section may be worth reading not only for people in the field.
