  We present a novel realisation of the $\mathbb{Z}_2\times\mathbb{Z}_2$-graded
Lie superalgebra $\mathfrak{gl}(m_1,m_2|n_1,n_2)$ inside an algebraic extension
of the enveloping algebra of the $\mathbb{Z}_2$-graded Lie superalgebra
$\mathfrak{gl}(m|n)$, with $m=m_1+m_2$ and $n=n_1+n_2$. A consequence of this
realisation is that the representations of $\mathfrak{gl}(m|n)$ "lift up" to
representations of $\mathfrak{gl}(m_1,m_2|n_1,n_2)$, with matrix elements
differing only by a sign, which we are able to characterise concisely.
