Define  3rd order commutator $R$ by $ R=[L_1,L_2]$. Then
\begin{small}
\[[L_j,R]=
A_1^{(j)}L_1^2+A_2^{(j)}L_2^2+A_3^{(j)}H^2+A_4^{(j)}\{L_1,L_2\}+A_5^{(j)}HL_1\]
\[+A_6^{(j)}HL_2 {+A_7^{(j)}L_1+A_8^{(j)}L_2+A_9^{(j)}H+A_{10}^{(j)}},\]
\[ R^2 = b_1 L_1^3 + b_2 L_2^3 + b_3 H^3 + b_4 \{L_1^2,L_2\} + b_5 \{L_1,L_2^2\}\]
\[ + b_6
L_1 L_2 L_1 + b_7 L_2 L_1 L_2
 + b_8 H\{L_1,L_2\} +b_9 H L_1^2 + b_{10} H L_2^2 \]
\[+ b_{11} H^2
L_1 + b_{12} H^2 L_2 + b_{13} L_1^2 + b_{14} L_2^2 + b_{15} \{L_1,L_2\} \]

\documentclass{article}

\usepackage{mathtools}

\begin{document}

\begin{gather*}
\begin{multlined}\relax
[L_j,R]=A_1^{(j)}L_1^2+A_2^{(j)}L_2^2+A_3^{(j)}H^2
\\+A_4^{(j)}\{L_1,L_2\}+A_5^{(j)}HL_1+A_6^{(j)}HL_2 +A_7^{(j)}L_1
\\+A_8^{(j)}L_2+A_9^{(j)}H+A_{10}^{(j)},
\end{multlined}
\\
\begin{multlined}
R^2 = b_1 L_1^3 + b_2 L_2^3 + b_3 H^3 + b_4 \{L_1^2,L_2\} + b_5 \{L_1,L_2^2\}
\\ + b_6 L_1 L_2 L_1 + b_7 L_2 L_1 L_2 + b_8 H\{L_1,L_2\}
\\ + b_9 H L_1^2 + b_{10} H L_2^2 + b_{11} H^2 L_1 + b_{12} H^2 L_2

\begin{align*}
[L_j,R]&=
A_1^{(j)}L_1^2+A_2^{(j)}L_2^2+A_3^{(j)}H^2 \\
&\qquad+A_4^{(j)}\{L_1,L_2\}+A_5^{(j)}HL_1+A_6^{(j)}HL_2 +A_7^{(j)}L_1 \\
&\qquad+A_8^{(j)}L_2+A_9^{(j)}H+A_{10}^{(j)}, \\[1ex]
R^2 &=
b_1 L_1^3 + b_2 L_2^3 + b_3 H^3 + b_4 \{L_1^2,L_2\} + b_5 \{L_1,L_2^2\} \\
&\qquad+ b_6 L_1 L_2 L_1 + b_7 L_2 L_1 L_2 + b_8 H\{L_1,L_2\} \\
&\qquad+ b_9 H L_1^2 + b_{10} H L_2^2 + b_{11} H^2 L_1 + b_{12} H^2 L_2 \\
&\qquad+ b_{13} L_1^2 + b_{14} L_2^2 + b_{15} \{L_1,L_2\} + b_{16} H L_1 \\
&\qquad+ b_{17} H L_2 + b_{18} H^2 + b_{19} L_1 + b_{20} L_2 + b_{21} H + b_{22},

The results  follow and generalize to all Laplace systems.
\begin{center}
\begin{figure}[H]
\centering
\includegraphics[scale=0.66]{commutingdiagram.png}
\caption{Relationship between conformal St\"ackel transforms and B\^ocher contractions}
\end{figure}

\end{center}
\begin{figure}[H]
\centering
\includegraphics[scale=0.107]{SchematicLaplaceHelmholtz2616.jpg}
\caption{The bigger picture}
\end{figure}

The Queen, who was accompanied by the Duke of Edinburgh during her walkabout in Windsor, was presented with a birthday cake at the Guildhall by the Great British Bake Off champion Nadiya Hussain, who had created an orange drizzle cake with a butter cream and marmalade filling.

The monarch unveiled a plaque marking The Queen's Walkway - a 6.3km trail that links 63 significant points in Windsor.

The trail was designed to recognise the moment the monarch broke the record on 9 September 2015 held by her great-great-grandmother Queen Victoria by being on the throne for 63 years and seven months.

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