Table 2 System algebras ${\mathfrak{t}}_{\mathit{M}}\mathbf{(}\mathit{L}\mathbf{)}$ of translation-invariant systems with $\mathbf{1}\mathbf{\le }\mathit{L}\mathbf{\le }\mathbf{6}$ spins and interaction lengths of less than M . Refer also to Theorem 25 for the structure of ${\mathfrak{t}}_{\mathit{L}}\mathbf{(}\mathit{L}\mathbf{)}$

1

$\mathfrak{su}(2)$

$\mathfrak{su}(2)$

$\mathfrak{su}(2)$

$\mathfrak{su}(2)$

2

$\mathfrak{su}(3)+\mathfrak{u}(1)$

$\mathfrak{su}(4)+{\sum }_{i=1}^2\mathfrak{su}(2)+\mathfrak{u}(1)$

$\mathfrak{su}(6)+\mathfrak{su}(4)+{\sum }_{i=1}^2\mathfrak{su}(2)+\mathfrak{u}(1)$

3

$\mathfrak{su}(4)+{\sum }_{i=1}^2\mathfrak{su}(2)+{\sum }_{i=1}^{\mathbf{2}}\mathfrak{u}(1)$

$\mathfrak{su}(6)+\mathfrak{su}(4)+{\sum }_{i=1}^2\mathfrak{su}(3)+{\sum }_{i=1}^{\mathbf{3}}\mathfrak{u}(1)$

4

$\mathfrak{su}(6)+\mathfrak{su}(4)+{\sum }_{i=1}^2\mathfrak{su}(3)+{\sum }_{i=1}^{\mathbf{3}}\mathfrak{u}(1)$

1

$\mathfrak{su}(2)$

$\mathfrak{su}(2)$

2

$\mathfrak{su}(8)+{\sum }_{i=1}^4\mathfrak{su}(6)+\mathfrak{u}(1)$

$\mathfrak{su}(14)+{\sum }_{i=1}^2\mathfrak{su}(11)+\mathfrak{su}(10)+{\sum }_{i=1}^2\mathfrak{su}(9)+\mathfrak{u}(1)$

3

$\mathfrak{su}(8)+{\sum }_{i=1}^4\mathfrak{su}(6)+{\sum }_{i=1}^{\mathbf{2}}\mathfrak{u}(1)$

$\mathfrak{su}(14)+{\sum }_{i=1}^2\mathfrak{su}(11)+\mathfrak{su}(10)+{\sum }_{i=1}^2\mathfrak{su}(9)+{\sum }_{i=1}^{\mathbf{3}}\mathfrak{u}(1)$

4

$\mathfrak{su}(8)+{\sum }_{i=1}^4\mathfrak{su}(6)+{\sum }_{i=1}^{\mathbf{4}}\mathfrak{u}(1)$

$\mathfrak{su}(14)+{\sum }_{i=1}^2\mathfrak{su}(11)+\mathfrak{su}(10)+{\sum }_{i=1}^2\mathfrak{su}(9)+{\sum }_{i=1}^{\mathbf{4}}\mathfrak{u}(1)$

5

$\mathfrak{su}(8)+{\sum }_{i=1}^4\mathfrak{su}(6)+{\sum }_{i=1}^{\mathbf{4}}\mathfrak{u}(1)$

$\mathfrak{su}(14)+{\sum }_{i=1}^2\mathfrak{su}(11)+\mathfrak{su}(10)+{\sum }_{i=1}^2\mathfrak{su}(9)+{\sum }_{i=1}^{\mathbf{5}}\mathfrak{u}(1)$

6

$\mathfrak{su}(14)+{\sum }_{i=1}^2\mathfrak{su}(11)+\mathfrak{su}(10)+{\sum }_{i=1}^2\mathfrak{su}(9)+{\sum }_{i=1}^{\mathbf{5}}\mathfrak{u}(1)$