@@ -1,155 +1,155 @@
-\documentstyle{article}
-\parindent0cm
-%\textwidth 15.5cm\textheight 22.0cm\columnwidth\textwidth
-%\hoffset-1.5cm\voffset-1.5cm
-\begin{document}
-%\parskip 10pt plus 1pt \parindent 0pt
-\title{The {LIE} Package}
-\author{Carsten and Franziska Sch\"obel\\
-The Leipzig University, Computer Science Dept.\\
-Augustusplatz 10/11, O-7010 Leipzig, Germany\\
-Email: cschoeb@aix550.informatik.uni-leipzig.de}
-\date{22 January 1993}
-\maketitle
-{\bf LIE} is a package of functions for the classification of real n-dimensional
-Lie algebras. It consists of two modules: {\bf liendmc1} and {\bf lie1234}.
-\\[0.3cm]{\large\bf liendmc1}\\[0.1cm]
-With the help of the functions in this module real n-dimensional Lie algebras
-$L$ with a derived algebra $L^{(1)}$ of dimension 1 can be classified. $L$ has
-to be defined by its structure constants $c_{ij}^k$ in the basis
-$\{X_1,\ldots,X_n\}$ with $[X_i,X_j]=c_{ij}^k X_k$. The user must define an
-ARRAY LIENSTRUCIN($n,n,n$) with n being the dimension of the Lie algebra $L$.
-The structure constants LIENSTRUCIN($i,j,k$):=$c_{ij}^k$ for $i<j$ should be
-given. Then the procedure LIENDIMCOM1 can be called. Its syntax is:
-\begin{verbatim}
-   LIENDIMCOM1(<number>).
-\end{verbatim}
-{\tt <number>} corresponds to the dimension $n$. The procedure simplifies
-the structure of $L$ performing real linear transformations. The returned
-value is a list of the form
-\begin{verbatim}
-   (i) {LIE_ALGEBRA(2),COMMUTATIVE(n-2)} or
-   (ii) {HEISENBERG(k),COMMUTATIVE(n-k)}
-\end{verbatim}
-with $3\leq k\leq n$, $k$ odd.\\
-The concepts correspond to the following theorem ({\tt LIE\_ALGEBRA(2)}
-$\rightarrow L_2$, {\tt HEISENBERG(k)} $\rightarrow H_k$ and
-{\tt COMMUTATIVE(n-k)} $\rightarrow C_{n-k}$):\\[0.2cm]
-{\bf Theorem.} Every real $n$-dimensional Lie algebra $L$ with a 1-dimensional
-derived algebra can be decomposed into one of the following forms:\\[0.1cm]
-\hspace*{0.3cm} (i) $C(L)\cap L^{(1)}=\{0\}\, :\; L_2\oplus C_{n-2}$
-or\\[0.05cm]
-\hspace*{0.3cm} (ii) $C(L)\cap L^{(1)}=L^{(1)}\, :\; H_k\oplus C_{n-k}\quad
-(k=2r-1,\, r\geq 2)$, with\newpage
-\hspace*{0.3cm} 1. $C(L)=C_j\oplus (L^{(1)}\cap C(L))$
-and dim$\,C_j=j$ ,\\[0.05cm]
-\hspace*{0.3cm} 2. $L_2$ is generated by
-$Y_1,Y_2$ with $[Y_1,Y_2]=Y_1$ ,\\[0.05cm]
-\hspace*{0.3cm} 3. $H_k$ is generated by $\{Y_1,\ldots,Y_k\}$ with\\
-\hspace*{0.7cm} $[Y_2,Y_3]=\cdots =[Y_{k-1},Y_k]=Y_1$.\\[0.2cm]
-(cf. \cite{cssmp92})\\[0.2cm]
-The returned list is also stored as LIE\_LIST. The matrix LIENTRANS gives the
-transformation from the given basis $\{X_1,\ldots ,X_n\}$ into the standard
-basis $\{Y_1,\ldots ,Y_n\}$: $Y_j=($LIENTRANS$)_j^k X_k$.\\[0.1cm]
-A more detailed output can be obtained by turning on the switch TR\_LIE:
-\begin{verbatim}
-   ON TR_LIE;
-\end{verbatim}
-before the procedure LIENDIMCOM1 is called.\\[0.1cm]
-The returned list could be an input for a data bank in which mathematical
-relevant properties of the obtained Lie algebras are stored.\\[0.3cm]
-{\large\bf lie1234}\\[0.1cm]
-This part of the package classifies real low-dimensional Lie algebras $L$
-of the dimension
-$n:=$dim$\,L=1,2,3,4$. $L$ is also given by its structure constants $c_{ij}^k$
-in the basis $\{X_1,\ldots,X_n\}$ with $[X_i,X_j]=c_{ij}^k X_k$. An ARRAY
-LIESTRIN($n,n,n$) has to be defined and LIESTRIN($i,j,k$):=$c_{ij}^k$ for
-$i<j$ should be given. Then the procedure LIECLASS can be performed
-whose syntax is:
-\begin{verbatim}
-   LIECLASS(<number>).
-\end{verbatim}
-{\tt <number>} should be the dimension of the Lie algebra $L$. The procedure
-stepwise simplifies the commutator relations of $L$ using properties of
-invariance like the dimension of the centre, of the derived algebra,
-unimodularity etc.  The returned value has the form:
-\begin{verbatim}
-   {LIEALG(n),COMTAB(m)},
-\end{verbatim}
-where $m$ corresponds to the number of the standard form (basis:
-$\{Y_1,\ldots,Y_n\}$) in an enumeration scheme. The corresponding enumeration
-schemes are listed below (cf. \cite{ntz-preprint27/92},\cite{mmpreprint1979}).
-In case that the standard form in the enumeration scheme depends on one (or two)
-parameter(s) $p_1$ (and $p_2$) the list is expanded to:
-\begin{verbatim}
-   {LIEALG(n),COMTAB(m),p1,p2}.
-\end{verbatim}
-This returned value is also stored as LIE\_CLASS. The linear transformation from
-the basis $\{X_1,\ldots,X_n\}$ into the basis of the standard form
-$\{Y_1,\ldots,Y_n\}$ is given by the matrix LIEMAT:
-$Y_j=($LIEMAT$)_j^k X_k$.\newpage
-By turning on the switch TR\_LIE:
-\begin{verbatim}
-   ON TR_LIE;
-\end{verbatim}
-before the procedure LIECLASS is called the output contains not only the
-list LIE\_CLASS but also the non-vanishing commutator relations in the
-standard form.\\[0.1cm]
-By the value $m$ and the parameters further examinations of the Lie algebra
-are possible, especially if in a data bank mathematical relevant properties
-of the enumerated standard forms are stored.\\[0.3cm]
-{\large\bf Enumeration schemes for lie1234}\\[0.2cm]
-\hspace*{0.3cm}\begin{tabular}{l|l}returned list LIE\_CLASS&
-the corresponding commutator relations\\[0.1cm]\hline
-{LIEALG(1),COMTAB(0)}&commutative case\\[0.1cm]\hline
-{LIEALG(2),COMTAB(0)}&commutative case\\[0.1cm]
-{LIEALG(2),COMTAB(1)}&$[Y_1,Y_2]=Y_2$\\[0.1cm]\hline
-{LIEALG(3),COMTAB(0)}&commutative case\\[0.1cm]
-{LIEALG(3),COMTAB(1)}&$[Y_1,Y_2]=Y_3$\\[0.1cm]
-{LIEALG(3),COMTAB(2)}&$[Y_1,Y_3]=Y_3$\\[0.1cm]
-{LIEALG(3),COMTAB(3)}&$[Y_1,Y_3]=Y_1,[Y_2,Y_3]=Y_2$\\[0.1cm]
-{LIEALG(3),COMTAB(4)}&$[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
-{LIEALG(3),COMTAB(5)}&$[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
-{LIEALG(3),COMTAB(6)}&$[Y_1,Y_3]=-Y_1+p_1 Y_2,[Y_2,Y_3]=Y_1,p_1\neq 0$\\[0.1cm]
-{LIEALG(3),COMTAB(7)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
-{LIEALG(3),COMTAB(8)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]\hline
-{LIEALG(4),COMTAB(0)}&commutative case\\[0.1cm]
-{LIEALG(4),COMTAB(1)}&$[Y_1,Y_4]=Y_1$\\[0.1cm]
-{LIEALG(4),COMTAB(2)}&$[Y_2,Y_4]=Y_1$\\[0,1cm]
-{LIEALG(4),COMTAB(3)}&$[Y_1,Y_3]=Y_1,[Y_2,Y_4]=Y_2$\\[0.1cm]
-{LIEALG(4),COMTAB(4)}&$[Y_1,Y_3]=-Y_2,[Y_2,Y_4]=Y_2,$\\
-                     &$[Y_1,Y_4]=[Y_2,Y_3]=Y_1$\\[0.1cm]
-{LIEALG(4),COMTAB(5)}&$[Y_2,Y_4]=Y_2,[Y_1,Y_4]=[Y_2,Y_3]=Y_1$\\[0.1cm]
-{LIEALG(4),COMTAB(6)}&$[Y_2,Y_4]=Y_1,[Y_3,Y_4]=Y_2$\\[0.1cm]
-{LIEALG(4),COMTAB(7)}&$[Y_2,Y_4]=Y_2,[Y_3,Y_4]=Y_1$\\[0.1cm]
-{LIEALG(4),COMTAB(8)}&$[Y_1,Y_4]=-Y_2,[Y_2,Y_4]=Y_1$\\[0.1cm]
-{LIEALG(4),COMTAB(9)}&$[Y_1,Y_4]=-Y_1+p_1 Y_2,[Y_2,Y_4]=Y_1,p_1\neq 0$\\[0.1cm]
-{LIEALG(4),COMTAB(10)}&$[Y_1,Y_4]=Y_1,[Y_2,Y_4]=Y_2$\\[0.1cm]
-{LIEALG(4),COMTAB(11)}&$[Y_1,Y_4]=Y_2,[Y_2,Y_4]=Y_1$
-\end{tabular}\\
-\hspace*{0.3cm}\begin{tabular}{l|l}returned list LIE\_CLASS&
-the corresponding commutator relations\\[0.1cm]\hline
-{LIEALG(4),COMTAB(12)}&$[Y_1,Y_4]=Y_1+Y_2,[Y_2,Y_4]=Y_2+Y_3,$\\
-                      &$[Y_3,Y_4]=Y_3$\\[0.1cm]
-{LIEALG(4),COMTAB(13)}&$[Y_1,Y_4]=Y_1,[Y_2,Y_4]=p_1 Y_2,[Y_3,Y_4]=p_2 Y_3,$\\
-                      &$p_1,p_2\neq 0$\\[0.1cm]
-{LIEALG(4),COMTAB(14)}&$[Y_1,Y_4]=p_1 Y_1+Y_2,[Y_2,Y_4]=-Y_1+p_1 Y_2,$\\
-                      &$[Y_3,Y_4]=p_2 Y_3,p_2\neq 0$\\[0.1cm]
-{LIEALG(4),COMTAB(15)}&$[Y_1,Y_4]=p_1 Y_1+Y_2,[Y_2,Y_4]=p_1 Y_2,$\\
-                      &$[Y_3,Y_4]=Y_3,p_1\neq 0$\\[0.1cm]
-{LIEALG(4),COMTAB(16)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\
-                      &$[Y_2,Y_4]=(1+p_1) Y_2,[Y_3,Y_4]=(1-p_1) Y_3,$\\
-                      &$p_1\geq 0$\\[0.1cm]
-{LIEALG(4),COMTAB(17)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\
-                      &$[Y_2,Y_4]=Y_2-p_1 Y_3,[Y_3,Y_4]=p_1 Y_2+Y_3,$\\
-                      &$p_1\neq 0$\\[0.1cm]
-{LIEALG(4),COMTAB(18)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\
-                      &$[Y_2,Y_4]=Y_2+Y_3,[Y_3,Y_4]=Y_3$\\[0.1cm]
-{LIEALG(4),COMTAB(19)}&$[Y_2,Y_3]=Y_1,[Y_2,Y_4]=Y_3,[Y_3,Y_4]=Y_2$\\[0.1cm]
-{LIEALG(4),COMTAB(20)}&$[Y_2,Y_3]=Y_1,[Y_2,Y_4]=-Y_3,[Y_3,Y_4]=Y_2$\\[0.1cm]
-{LIEALG(4),COMTAB(21)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
-{LIEALG(4),COMTAB(22)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$
-\end{tabular}
-\bibliography{lie}
-\bibliographystyle{plain}
-\end{document}
+\documentstyle{article}
+\parindent0cm
+%\textwidth 15.5cm\textheight 22.0cm\columnwidth\textwidth
+%\hoffset-1.5cm\voffset-1.5cm
+\begin{document}
+%\parskip 10pt plus 1pt \parindent 0pt
+\title{The {LIE} Package}
+\author{Carsten and Franziska Sch\"obel\\
+The Leipzig University, Computer Science Dept.\\
+Augustusplatz 10/11, O-7010 Leipzig, Germany\\
+Email: cschoeb@aix550.informatik.uni-leipzig.de}
+\date{22 January 1993}
+\maketitle
+{\bf LIE} is a package of functions for the classification of real n-dimensional
+Lie algebras. It consists of two modules: {\bf liendmc1} and {\bf lie1234}.
+\\[0.3cm]{\large\bf liendmc1}\\[0.1cm]
+With the help of the functions in this module real n-dimensional Lie algebras
+$L$ with a derived algebra $L^{(1)}$ of dimension 1 can be classified. $L$ has
+to be defined by its structure constants $c_{ij}^k$ in the basis
+$\{X_1,\ldots,X_n\}$ with $[X_i,X_j]=c_{ij}^k X_k$. The user must define an
+ARRAY LIENSTRUCIN($n,n,n$) with n being the dimension of the Lie algebra $L$.
+The structure constants LIENSTRUCIN($i,j,k$):=$c_{ij}^k$ for $i<j$ should be
+given. Then the procedure LIENDIMCOM1 can be called. Its syntax is:
+\begin{verbatim}
+   LIENDIMCOM1(<number>).
+\end{verbatim}
+{\tt <number>} corresponds to the dimension $n$. The procedure simplifies
+the structure of $L$ performing real linear transformations. The returned
+value is a list of the form
+\begin{verbatim}
+   (i) {LIE_ALGEBRA(2),COMMUTATIVE(n-2)} or
+   (ii) {HEISENBERG(k),COMMUTATIVE(n-k)}
+\end{verbatim}
+with $3\leq k\leq n$, $k$ odd.\\
+The concepts correspond to the following theorem ({\tt LIE\_ALGEBRA(2)}
+$\rightarrow L_2$, {\tt HEISENBERG(k)} $\rightarrow H_k$ and
+{\tt COMMUTATIVE(n-k)} $\rightarrow C_{n-k}$):\\[0.2cm]
+{\bf Theorem.} Every real $n$-dimensional Lie algebra $L$ with a 1-dimensional
+derived algebra can be decomposed into one of the following forms:\\[0.1cm]
+\hspace*{0.3cm} (i) $C(L)\cap L^{(1)}=\{0\}\, :\; L_2\oplus C_{n-2}$
+or\\[0.05cm]
+\hspace*{0.3cm} (ii) $C(L)\cap L^{(1)}=L^{(1)}\, :\; H_k\oplus C_{n-k}\quad
+(k=2r-1,\, r\geq 2)$, with\newpage
+\hspace*{0.3cm} 1. $C(L)=C_j\oplus (L^{(1)}\cap C(L))$
+and dim$\,C_j=j$ ,\\[0.05cm]
+\hspace*{0.3cm} 2. $L_2$ is generated by
+$Y_1,Y_2$ with $[Y_1,Y_2]=Y_1$ ,\\[0.05cm]
+\hspace*{0.3cm} 3. $H_k$ is generated by $\{Y_1,\ldots,Y_k\}$ with\\
+\hspace*{0.7cm} $[Y_2,Y_3]=\cdots =[Y_{k-1},Y_k]=Y_1$.\\[0.2cm]
+(cf. \cite{cssmp92})\\[0.2cm]
+The returned list is also stored as LIE\_LIST. The matrix LIENTRANS gives the
+transformation from the given basis $\{X_1,\ldots ,X_n\}$ into the standard
+basis $\{Y_1,\ldots ,Y_n\}$: $Y_j=($LIENTRANS$)_j^k X_k$.\\[0.1cm]
+A more detailed output can be obtained by turning on the switch TR\_LIE:
+\begin{verbatim}
+   ON TR_LIE;
+\end{verbatim}
+before the procedure LIENDIMCOM1 is called.\\[0.1cm]
+The returned list could be an input for a data bank in which mathematical
+relevant properties of the obtained Lie algebras are stored.\\[0.3cm]
+{\large\bf lie1234}\\[0.1cm]
+This part of the package classifies real low-dimensional Lie algebras $L$
+of the dimension
+$n:=$dim$\,L=1,2,3,4$. $L$ is also given by its structure constants $c_{ij}^k$
+in the basis $\{X_1,\ldots,X_n\}$ with $[X_i,X_j]=c_{ij}^k X_k$. An ARRAY
+LIESTRIN($n,n,n$) has to be defined and LIESTRIN($i,j,k$):=$c_{ij}^k$ for
+$i<j$ should be given. Then the procedure LIECLASS can be performed
+whose syntax is:
+\begin{verbatim}
+   LIECLASS(<number>).
+\end{verbatim}
+{\tt <number>} should be the dimension of the Lie algebra $L$. The procedure
+stepwise simplifies the commutator relations of $L$ using properties of
+invariance like the dimension of the centre, of the derived algebra,
+unimodularity etc.  The returned value has the form:
+\begin{verbatim}
+   {LIEALG(n),COMTAB(m)},
+\end{verbatim}
+where $m$ corresponds to the number of the standard form (basis:
+$\{Y_1,\ldots,Y_n\}$) in an enumeration scheme. The corresponding enumeration
+schemes are listed below (cf. \cite{ntz-preprint27/92},\cite{mmpreprint1979}).
+In case that the standard form in the enumeration scheme depends on one (or two)
+parameter(s) $p_1$ (and $p_2$) the list is expanded to:
+\begin{verbatim}
+   {LIEALG(n),COMTAB(m),p1,p2}.
+\end{verbatim}
+This returned value is also stored as LIE\_CLASS. The linear transformation from
+the basis $\{X_1,\ldots,X_n\}$ into the basis of the standard form
+$\{Y_1,\ldots,Y_n\}$ is given by the matrix LIEMAT:
+$Y_j=($LIEMAT$)_j^k X_k$.\newpage
+By turning on the switch TR\_LIE:
+\begin{verbatim}
+   ON TR_LIE;
+\end{verbatim}
+before the procedure LIECLASS is called the output contains not only the
+list LIE\_CLASS but also the non-vanishing commutator relations in the
+standard form.\\[0.1cm]
+By the value $m$ and the parameters further examinations of the Lie algebra
+are possible, especially if in a data bank mathematical relevant properties
+of the enumerated standard forms are stored.\\[0.3cm]
+{\large\bf Enumeration schemes for lie1234}\\[0.2cm]
+\hspace*{0.3cm}\begin{tabular}{l|l}returned list LIE\_CLASS&
+the corresponding commutator relations\\[0.1cm]\hline
+{LIEALG(1),COMTAB(0)}&commutative case\\[0.1cm]\hline
+{LIEALG(2),COMTAB(0)}&commutative case\\[0.1cm]
+{LIEALG(2),COMTAB(1)}&$[Y_1,Y_2]=Y_2$\\[0.1cm]\hline
+{LIEALG(3),COMTAB(0)}&commutative case\\[0.1cm]
+{LIEALG(3),COMTAB(1)}&$[Y_1,Y_2]=Y_3$\\[0.1cm]
+{LIEALG(3),COMTAB(2)}&$[Y_1,Y_3]=Y_3$\\[0.1cm]
+{LIEALG(3),COMTAB(3)}&$[Y_1,Y_3]=Y_1,[Y_2,Y_3]=Y_2$\\[0.1cm]
+{LIEALG(3),COMTAB(4)}&$[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
+{LIEALG(3),COMTAB(5)}&$[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
+{LIEALG(3),COMTAB(6)}&$[Y_1,Y_3]=-Y_1+p_1 Y_2,[Y_2,Y_3]=Y_1,p_1\neq 0$\\[0.1cm]
+{LIEALG(3),COMTAB(7)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
+{LIEALG(3),COMTAB(8)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]\hline
+{LIEALG(4),COMTAB(0)}&commutative case\\[0.1cm]
+{LIEALG(4),COMTAB(1)}&$[Y_1,Y_4]=Y_1$\\[0.1cm]
+{LIEALG(4),COMTAB(2)}&$[Y_2,Y_4]=Y_1$\\[0,1cm]
+{LIEALG(4),COMTAB(3)}&$[Y_1,Y_3]=Y_1,[Y_2,Y_4]=Y_2$\\[0.1cm]
+{LIEALG(4),COMTAB(4)}&$[Y_1,Y_3]=-Y_2,[Y_2,Y_4]=Y_2,$\\
+                     &$[Y_1,Y_4]=[Y_2,Y_3]=Y_1$\\[0.1cm]
+{LIEALG(4),COMTAB(5)}&$[Y_2,Y_4]=Y_2,[Y_1,Y_4]=[Y_2,Y_3]=Y_1$\\[0.1cm]
+{LIEALG(4),COMTAB(6)}&$[Y_2,Y_4]=Y_1,[Y_3,Y_4]=Y_2$\\[0.1cm]
+{LIEALG(4),COMTAB(7)}&$[Y_2,Y_4]=Y_2,[Y_3,Y_4]=Y_1$\\[0.1cm]
+{LIEALG(4),COMTAB(8)}&$[Y_1,Y_4]=-Y_2,[Y_2,Y_4]=Y_1$\\[0.1cm]
+{LIEALG(4),COMTAB(9)}&$[Y_1,Y_4]=-Y_1+p_1 Y_2,[Y_2,Y_4]=Y_1,p_1\neq 0$\\[0.1cm]
+{LIEALG(4),COMTAB(10)}&$[Y_1,Y_4]=Y_1,[Y_2,Y_4]=Y_2$\\[0.1cm]
+{LIEALG(4),COMTAB(11)}&$[Y_1,Y_4]=Y_2,[Y_2,Y_4]=Y_1$
+\end{tabular}\\
+\hspace*{0.3cm}\begin{tabular}{l|l}returned list LIE\_CLASS&
+the corresponding commutator relations\\[0.1cm]\hline
+{LIEALG(4),COMTAB(12)}&$[Y_1,Y_4]=Y_1+Y_2,[Y_2,Y_4]=Y_2+Y_3,$\\
+                      &$[Y_3,Y_4]=Y_3$\\[0.1cm]
+{LIEALG(4),COMTAB(13)}&$[Y_1,Y_4]=Y_1,[Y_2,Y_4]=p_1 Y_2,[Y_3,Y_4]=p_2 Y_3,$\\
+                      &$p_1,p_2\neq 0$\\[0.1cm]
+{LIEALG(4),COMTAB(14)}&$[Y_1,Y_4]=p_1 Y_1+Y_2,[Y_2,Y_4]=-Y_1+p_1 Y_2,$\\
+                      &$[Y_3,Y_4]=p_2 Y_3,p_2\neq 0$\\[0.1cm]
+{LIEALG(4),COMTAB(15)}&$[Y_1,Y_4]=p_1 Y_1+Y_2,[Y_2,Y_4]=p_1 Y_2,$\\
+                      &$[Y_3,Y_4]=Y_3,p_1\neq 0$\\[0.1cm]
+{LIEALG(4),COMTAB(16)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\
+                      &$[Y_2,Y_4]=(1+p_1) Y_2,[Y_3,Y_4]=(1-p_1) Y_3,$\\
+                      &$p_1\geq 0$\\[0.1cm]
+{LIEALG(4),COMTAB(17)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\
+                      &$[Y_2,Y_4]=Y_2-p_1 Y_3,[Y_3,Y_4]=p_1 Y_2+Y_3,$\\
+                      &$p_1\neq 0$\\[0.1cm]
+{LIEALG(4),COMTAB(18)}&$[Y_1,Y_4]=2 Y_1,[Y_2,Y_3]=Y_1,$\\
+                      &$[Y_2,Y_4]=Y_2+Y_3,[Y_3,Y_4]=Y_3$\\[0.1cm]
+{LIEALG(4),COMTAB(19)}&$[Y_2,Y_3]=Y_1,[Y_2,Y_4]=Y_3,[Y_3,Y_4]=Y_2$\\[0.1cm]
+{LIEALG(4),COMTAB(20)}&$[Y_2,Y_3]=Y_1,[Y_2,Y_4]=-Y_3,[Y_3,Y_4]=Y_2$\\[0.1cm]
+{LIEALG(4),COMTAB(21)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=-Y_2,[Y_2,Y_3]=Y_1$\\[0.1cm]
+{LIEALG(4),COMTAB(22)}&$[Y_1,Y_2]=Y_3,[Y_1,Y_3]=Y_2,[Y_2,Y_3]=Y_1$
+\end{tabular}
+\bibliography{lie}
+\bibliographystyle{plain}
+\end{document}