  In the framework of operator theory, we investigate a close Lie theoretic
relationship between all operator ideals and certain classical groups of
invertible operators that can be described as the solution sets of certain
algebraic equations, hence can be regarded as infinite-dimensional linear
algebraic groups. Historically, this has already been done for only the
complete-norm ideals; in that case one can work within the framework of the
well-known Lie theory for Banach-Lie groups. That kind of Lie theory is not
applicable for arbitrary operator ideals, so we needed to find a new approach
for dealing with the general situation. The simplest instance of the
aforementioned relationship is provided by the Lie algebra
$\ug_{\Ic}(\Hc)=\{X\in\Ic\mid X^*=-X\}$ associated with the group
$\U_{\Ic}(\Hc)=\U(\Hc)\cap(\1+\Ic)$ where $\Ic$ is an arbitrary operator ideal
in $\Bc(\Hc)$ and $\U(\Hc)$ is the full group of unitary operators. We
investigate the Cartan subalgebras (maximal abelian self-adjoint subalgebras)
of $\ug_{\Ic}(\Hc)$ for $\{0\}\subsetneqq\Ic\subsetneqq\Bc(\Hc)$, and obtain an
uncountably many $\U_{\Ic}(\Hc)$-conjugacy classes of these Cartan subalgebras.
The cardinality proof will be given in a follow up paper \cite{BPW13} and
stands in contrast to the $\U(\Hc)$-uniqueness work of de la Harpe
\cite{dlH72}.
