Express a vector field in terms of a frame

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I have a set of 4 vector fields that form a frame, I have defined them as

N = Manifold(4, 'N')
Y.<x,y,k,t> = N.chart()
F = Y.frame()
X_1 = cos(k)*F[0] + sin(k)*F[1] + (-y/2*cos(k) + x/2*sin(k))*F[3]
X_2 = -sin(k)*F[0] + cos(k)*F[1] + (x/2*cos(k) + y/2*sin(k))*F[3]
X_3 = F[3]
b = var('b', domain='real') 
assume(2 > b, b >= 0) 
X_4 = F[2] - b*F[3]

Now I want to calculate their Lie bracket and express it in terms of these vectors fields, for example the following gives [X_2, X_4], and I would like that written in terms of the {X_1, X_2, X_3, X_4}, in this case the output should be X_1.

vw = X_2.bracket(X_4)
vw
vw.display()

In general, I would like to know, given a vector field V in the original frame, how it can be expressed in terms of {X_1, X_2, X_3, X_4}. Many thanks.