Similarity Group
Representation
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}

simMatrix = Table[0,{i,4},{j,4}]
simMatrix[[1;;3,1;;3]] = SkewSym[v]
simMatrix[[1;;3,4]] = u
simMatrix[[4,4]] = lambda
MatrixForm[simMatrix]
(*simExponetialMap = Series[simMatrix,{u,{0,0,0},2}]*)

Exponential map, refer to Ethan Eade
v_skew = SkewSym(v)
theta =sqrt(Transpose(v).v)
X = sin(theta)/theta
Y = (1-cos(theta))/(theta^2)
Z = (1-X)/(theta^2)
W = (0.5 - Y)/(theta^2)
alpha = lambda^2 / (lambda^2 + theta^2)
beta = (exp(-lambda)-1+lambda)/(lambda^2)
gamma = Y - lambda.Z
mu = (1-lambda + 0.5*lambda^2 - exp(-lambda))/(lambda^2)
nu = Z-lambda W
Av = (1-exp(-lambda))/lambda
Bv = alpha .(beta - lambda) + lambda
Cv = alpha .(mu - nv) + nv
a = sin(theta)/theta
b = (1-cos(theta))/(theta^2)
R = IdentityMatrix[3]+ a.v_skew + b.v_skew.v_skew
V = Av.IdentityMatrix[3]+Bv.v_skew + Cv. v_skew.v_skew
simExponetialMap = Table[0,{i,4},{j,4}]
simExponetialMap[[1;;3,1;;3]] = R
simExponetialMap[[1;;3,4]] = V.u
simExponetialMap[[4,4]] = exp(-lambda)
MatrixForm[simExponetialMap]

f[lambda_] = simExponetialMap
f'[lambda]