Similarity Group
Representation
A similarity transformation is a combination 
Exponential and Logmap

Retract and localCoordinate

Lie group generators for similarity transform:
G1 = {{0,0,0,1},{0,0,0,0},{0,0,0,0},{0,0,0,0}}
G2 = {{0,0,0,0},{0,0,0,1},{0,0,0,0},{0,0,0,0}}
G3 = {{0,0,0,0},{0,0,0,0},{0,0,0,1},{0,0,0,0}}
G4 = {{0,0,0,0},{0,0,-1,0},{0,1,0,0},{0,0,0,0}}
G5 = {{0,0,1,0},{0,0,0,0},{-1,0,0,0},{0,0,0,0}}
G6 = {{0,-1,0,0},{1,0,0,0},{0,0,0,1},{0,0,0,0}}
G7 = {{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,-1}}

Lie vectors similarity3 can be described as:
u = {x,y,z}
v = {w1,w2,w3}
simLieVector = {u,v,lambda}

The matrix form of the similarity vector is:
simMatrix = Table[0,{i,4},{j,4}]
simMatrix[[1;;3,1;;3]] = SkewSym[v]

The exponential map of similarity3 is:

simExponetialMap = Expand[Series[]]

I3 = IdentyMatrix[3]