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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()

X_1 = cos(k) Y.frame()[0] + sin(k)Y.frame()[1] +(-y/2 cos(k) + x/2 *sin(k) )Y.frame()[3]

X_2 = -sin(k) Y.frame()[0] + cos(k)Y.frame()[1] +(x/2 cos(k) + y/2 *sin(k) )Y.frame()[3]

X_3 = Y.frame()[3]

b = var('b', domain='real')

assume(2>b, b>=0)

X_4 = Y.frame()[2]-b* Y.frame()[3]

Now I want to calculate the Lie bracket of them and express it in terms of these vectors fields, for example the following gives [X_2,X_4], and I would like that the return is 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, how given a vector field V in the original frame, can be expressed in terms of {X_1,X_2,X_3,X_4}. Many thanks

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

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

assume(2 > b, b >= 0) 
X_4 = Y.frame()[2]-b* Y.frame()[3]Y.frame()[2] - b* Y.frame()[3]

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

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

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) *Y.frame()[0] cos(k)*F[0] + sin(k)*Y.frame()[1] +(-y/2 *cos(k) sin(k)*F[1] + x/2 *sin(k) )*Y.frame()[3]
(-y/2*cos(k) + x/2*sin(k))*F[3]
X_2 = -sin(k) *Y.frame()[0] -sin(k)*F[0] + cos(k)*Y.frame()[1] +(x/2 *cos(k) cos(k)*F[1] + y/2 *sin(k) )*Y.frame()[3]
(x/2*cos(k) + y/2*sin(k))*F[3]
X_3 = Y.frame()[3]
F[3]
b = var('b', domain='real') 
assume(2 > b, b >= 0) 
X_4 = Y.frame()[2] F[2] - b* Y.frame()[3]
b*F[3]

Now I want to calculate the their Lie bracket of them and express it in terms terms of these vectors fields, fields, for example the following gives [X_2, X_4], and I would like that the return is written 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); 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.