The ‘right trivialised’ tangent of the exponential map, dexpR, according to Iserles05an, formula 2.42, pg. 32 can be written as
	({{	}, {g'(g^(-1))=(Subscript[dexpR, ω])(ω')}})
where g=exp(ω), and g'=exp'(ω).
Compare this to the left Jacobian matrix (Subscript[J, l]) in Chirikjian11book2, pg. 26, we see that dexpR and (Subscript[J, l]) are the same.

Hence, be careful, Iserles’s ‘right trivialised’ tangent of the exponential map dexpR is in fact Chirikjian’s left Jacobian matrix (Subscript[J, l])!!!

We want to compute the s ‘left trivialised’ tangent of the exponential map, dexpL, for SE2, hence, we need to use Chirikjian’s right Jacobian matrix (Subscript[J, r]) formula in Chirikjian11book2, pg. 36.

Clear[J]
J[α_]:={{Sin[α]/α,(1-Cos[α])/α, (α (Subscript[v, 1])-(Subscript[v, 2])+(Subscript[v, 2])Cos[α]-(Subscript[v, 1])Sin[α])/(α^2)},{-(1-Cos[α])/α,Sin[α]/α, ((Subscript[v, 1])+α (Subscript[v, 2])-(Subscript[v, 1])Cos[α]-(Subscript[v, 2])Sin[α])/(α^2)},{0,0,1}}

Jinv[α_]=Inverse[J[α]]

Jinv[α]//Simplify//MatrixForm

In case α=0, we compute the limits of (Subscript[J, r]) and ((Subscript[J, r])^(-1)) as follows

Limit[Jinv[α],α->0]//Simplify//MatrixForm

Limit[J[α],α->0]//Simplify//MatrixForm