Let
$V$
be a vector space over a field
$\mathbb{K}$
of characteristic zero and
${{V}_{*}}$
be a space of linear functionals on
$V$
which separate the points of
$V$
. We consider
$V\,\otimes \,{{V}_{*}}$
as a Lie algebra of finite rank operators on
$V$
, and set
$\mathfrak{g}\mathfrak{l}(V,\,{{V}_{*}})\,:=\,V\,\otimes \,{{V}_{*}}$
. We define a Cartan subalgebra of
$\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$
as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of
$\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$
under the assumption that
$\mathbb{K}$
is algebraically closed. A subalgebra of
$\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$
is a Cartan subalgebra if and only if it equals
${{\oplus }_{j}}({{V}_{j}}\,\otimes {{({{V}_{j}})}_{*}})\,\oplus \,({{V}^{0}}\,\otimes \,V_{*}^{0})$
for some one-dimensional subspaces
${{V}_{j}}\subseteq V$
and
${{\text{(}{{V}_{j}}\text{)}}_{*}}\subseteq {{V}_{*}}$
with
${{({{V}_{i}})}_{*}}({{V}_{j}})\,=\,{{\delta }_{ij}}\mathbb{K}$
and such that the spaces
$V_{*}^{0}=\bigcap{_{j}}{{({{V}_{j}})}^{\bot }}\subseteq {{V}_{*}}$
and
${{V}^{0}}=\bigcap{_{j}}{{\left( {{({{V}_{j}})}_{*}} \right)}^{\bot }}\subseteq V$
satisfy
$V_{*}^{0}({{V}^{0}})\,=\,\{0\}$
. We then discuss explicit constructions of subspaces
${{V}_{j}}$
and
${{({{V}_{j}})}_{*}}$
as above. Our second main result claims that a Cartan subalgebra of
$\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$
can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra
$\mathfrak{h}$
which coincides with the maximal locally nilpotent
$\mathfrak{h}$
-submodule of
$\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$
, and such that the adjoint representation of
$\mathfrak{h}$
is locally finite.