The number of different lattice types is limited. The following observations can explain the statement

1) A unit cell with two centered faces must be of type F.

    Suppose the cell is A and B  centred. Then a lattice row defined by the lattice points (0, 1/2, 1/2) and (1/2,0,1/2) occurs. Since any lattice point is equivalent to any other because of the translation symmetry, a parallel lattice row with the same period should pass through the lattice point (1,0,0), which involves an additional lattice point at (1/2,1/2,0). The cell is then F-centred.

2) A unit cell which is at same time I- and C-centred can always be reduced to a conventional centred cell. 

    For instance an I and C cell will have lattice points at position (1/2,1/2,1/2)  and (1/2,1/2,0). A  lattice row arises passing through  the two lattice points. In accordance with the previous case, a lattice point at (0,0,1/2) should be present. The lattice can then be described by a new A cell with axes a' = a, b' = b, c' = c/2

Triclinic lattices

    Even though non-primitive cells can always be chosen, the absence of axes with order greater than one suggests the choice of a conventional primitive cell with unrestricted , , angles and a:b:c ratios. In fact, any triclinic lattice can always be referred to such a cell.

Monoclinic lattices

    The  conventional cell has the twofold axis parallel to b,  angles = = 90°, unrestricted b and a:b:c ratios. Let us consider first a B-centred monoclinic cell with unit vectors a, b, c. In the figure we draw two adjacent cells. We can choose a new primitive cell with parameters:

 a'  = a,  b'  =  b,   c'  = (a + c)  / 2               

Since c' lies on the (a, c) plane the new cell will be still monoclinic. Therefore a lattice with a B type monoclinic cell can always be reduced to a lattice with P monoclinic cell. 

    Let us now consider an I unit cell with axes a, b, c; In the figure we show two adjacent cells. A new  A-centred cell may be choosen with parameters:  

 a' = a,  b' = b,    c' = a + c          

Therefore a lattice with an I unit monoclinic cell may always be described by an A monoclinic cell. Furthermore, since the a and c axes can always be interchanged, an A cell can be always reduced to a C cell.

    A lattice described by an F-monoclinic cell can always be described by a C-monoclinic cell by choosing

a'  =  a,    b'  =  b,    c' = (a + c) / 2.

    A lattice with a C-monoclinic cell is not amenable to a lattice having a P monoclinic cell. By a C cell with axes a, b, c a primitive cell is obtained by assuming

a' = (a + b) / 2,  b' = (-a + b) / 2,   c' = c

but this no longer shows the features of a monoclinic cell since     90°, a'=b' c' and the two axes lies along the diagonal of the face. It can be concluded that there are two distinct monoclinic lattices described by P and C cells, and not amenable one on the other.      

Orthorhombic lattices

    In the conventional orthorhombic cell the three proper or inversion axes are parallel to the vectors a, b, c, with angles = = = 90° and general a:b:c ratios.

    An A-centred or a B-centred unit cell can be reduced to a C-centred orthorhombic cell just by  trivially interchanging the axes.

    An I-centred cell cannot be reduced to a P  or to a C-centred orthorhombic cell. In order to obtain a P unit cell from an I-cell the following axis transformation may be used (in the figure, eight adjacent cells are shown):

a' = - ½ a + ½ b  + ½ c

b' =   ½ a  -  ½ b + ½ c                

c' =   ½ a  + ½ b  - ½ c

which is not a conventional orthorhombic unit cell. 

In order to show that an I-centred cell cannot be reduced to a C-centred conventionally orthorhombic cell, the user can apply the following axis transformation:

a'  =  a

b'  =  b + c              

c'  =  c

    An F-centred cell cannot be reduced to a conventional orthorhombic P- or C- or I-centred cell. In order to obtain from an F-centred a P-unit cell in following axis - transformation can be used:

a'  =   ½ b  +  ½ c

b'  =   ½ a  +  ½ c        

c'  =   ½ a  +  ½ b

The resulting P-unit cell is not a conventional orthorhombic cell, thus the F-lattice is different from a P- lattice.

    The user can verify that an F-centred cell cannot be reduced to a conventional C-cell by using the following axis transformation:

a'  =   a

b'  =   ½ b  +  ½ c        

c'  =   c 

Tetragonal lattices

    In the conventional tetragonal cell the fourfold axis is chosen along c with = = = 90°, a = b and unrestricted c value

    Because of the 4-fold symmetry axis, a tetragonal A-centred cell is also a B-cell, and therefore an F-centred cell. This last cell can be reduced to a conventional tetragonal I-cell by the following axis transformation:

a'  =   ½ a  +  ½ b

b'  =  - ½ a  +  ½ b         

c'  =   c

A C cell is always amenable to a conventional tetragonal P-cell by the following axis - transformation:

a'  =    ½ a + ½ b

b'  =  - ½ a + ½ b         

c'   =   c

Thus only two different tetragonal lattices, P and I, are found.

Cubic lattices

    In the conventional cubic cell, the four threefold axes are chosen to be parallel to the principal diagonals of a cube, while the unit vectors a, b, c are parallel to the cube edges.

There are only P,  I, and F cubic lattices.

Suppose you have an A-lattice. Because of the 3-fold symmetry axes the A-lattice should also be a B-lattice and therefore an F-lattice. You can go from such a lattice to a new P-lattice by choosing

a' =  ½ c  +  ½ b 

b' =  ½ a  +  ½ c           

c'=   ½ a  +  ½ b

This new primitive cell does not correspond to a conventional cubic cell, and therefore corresponds to a new type of lattice, the F-centred cubic lattice.

Suppose you have an I-lattice. You can go from an I  to a P-lattice by choosing

a' =  -½ a  +  ½ b  + ½ c

b' =   ½ a  -  ½ c  +  ½ c        

c'=    ½ a  +  ½ b  -  ½ c

This new primitive cell does not correspond to a conventional cubic cell, and therefore corresponds to a new type of lattice, the I-centred lattice.

Hexagonal lattices

In the conventional hexagonal cell the sixfold axis is chosen parallel to c, with a = b, unrestricted c, = = 90°, and = 120°. Only P lattices can exist. Suppose we have a A- unit cell. Apply the 6-fold symmetry: then the A-lattice must be B (and therefore F) and I-lattice simultaneously. The result: an hexagonal A- lattice may be reduced to an hexagonal P-lattice with axes

a' = ½ a    b' = ½ b    c'=½ c

In an analogous way one can start from a B-lattice or a C-lattice; any of them can be reduced to a P lattice.

Trigonal lattices

    As in the hexagonal cell, so in the conventional trigonal cell the threefold axis is chosen parallel to  c, with a = b,  = = 90°,  and = 120°. As in the hexagonal case, A-, B-, C-, F- centred cells are easily amenable to conventional P trigonal cell. In all these cases lattice planes normal to c completely overlap when projected along c.

Rhombohedral lattices also exist. They occur when, between two adjacent planes considered for the P case, two supplementary planes exist shifted by (2/3,1/3,1/3) and by (1/3,2/3,2/3) respectively. In this case a rhombohedral primitive cell  can be chosen.