\( \DeclareMathOperator{\op}{op} \DeclareMathOperator{\ad}{ad} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Lie}{Lie} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\img}{im} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Real}{Re} \DeclareMathOperator{\Imag}{Im} \DeclareMathOperator{\adj}{adj} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Cl}{Cl} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\MM}{\mathbb{M}} \newcommand{\mapto}{\rightarrow} \newcommand\abs[1]{\left|#1\right|} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\dagg}{\dagger} \def\bra#1{\mathinner{\langle{#1}|}} \def\ket#1{\mathinner{|{#1}\rangle}} \def\braket#1{\mathinner{\langle{#1}\rangle}} \def\Bra#1{\left\langle#1\right|} \)