The following is based on Lévy-LeBlond’s *Galilei Group and Galilean Invariance*, §2 (*Nuovo Cimento*, Jan. 1973).

Let *Ω* be the complete Newtonian space, the points (events) of which we label by their coordinates in some complete Galilean frame, using the notation

*y* = (**x**(*t*), **z**(*s*)). (1)

The complete proper Galilei group *G* (or Galilei group for short) is a group of linear endomorphisms of *Ω*, of the form

*y′* = (**x′**(*t*), **z′**(*s*))

which are

*t′* = *t* + *g*, (2a)

*s′* = *s* + *h*, (2b)

**x′** = *R***x** + **v***t* + **a**, (2c)

**z′** = *P***z** + **w***s* + **b**, (2d)

where *R* and *P* are 3 × 3 orthogonal matrices. The physical interpretation is straight-forward: The point *y′* is obtained from *y* by a time translation *g*, a freelength translation *h*, a length-space translation **a**, a duration-space translation **b**, a length-space Galilean transformation **v**, a duration-space Galilean transformation **w**, a length-space rotation *R* and a duration-space rotation *P*.

The length-space Galilean transformation consists in a uniform motion with velocity **v**. The duration-space Galilean transformation consists in a uniform motion with lenticity **w**. It is important to notice that a length-space Galilean transformation is effected at a very definite instant of time, *t* = 0, and a duration-space Galilean transformation is effected at a very definite point of freelength, *s* = 0, in the parameterization of (2).

Instead of this active point of view, one can adopt the equivalent passive point of view; (2) would then give the relation between the coordinates of the same event as described in two different complete Galilean frames.

The complete Galilei group thus appears as a subgroup of the affine group. Under such a general Galilean transformation, the time interval between any two events is invariant,

*t*_{2} − *t*_{1} = const, (3)

and the freelength interval between any two events is invariant,

*s*_{2} − *s*_{1} = const. (4)

This also applies to the length-space distance between *simultaneous* events,

|**x**_{2} − **x**_{1}| = const if *t*_{2} = *t*_{1}. (5)

and the length-space distance between *simulindistant* events,

|**z**_{2} − **z**_{1}| = const if *s*_{2} = *s*_{1}. (6)

Conversely, the complete Galilei group is the most general *linear* group of transformations in *Ω*, such that (3) to (6) hold true. The restriction to linear transformations is crucial since otherwise any “rigid transformation”

*t′* = *t* + *g*, *R***x** + **f**(*t*), (7)

where **f** is an arbitrary function of time, obeys (3) and (5). Similarly, without the restriction to linear transformations, any “rigid transformation”

*s′* = *s* + *h*, *P***z** + **k**(*s*), (8)

where **k** is an arbitrary function of freelength, obeys (4) and (6).

The complete Galilei group *G* also has as a subgroup the Euclidean group *E* of length-space displacements and duration-space dischronments.