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It suffices to attach the function mybracket that you have defined to the vector field class, which is VectorField. You have to import the latter first:

from sage.manifolds.differentiable.vectorfield import VectorField
VectorField.mybracket = mybracket

So the whole code becomes:

from sage.manifolds.differentiable.vectorfield import VectorField
def mybracket(A, B):
     return B.bracket(A)
VectorField.mybracket = mybracket

Then

M.<x,y> = EuclideanSpace()
X = M.vector_field(0,1-y*x^2, name='X')
Y = M.vector_field(-(y-1),x, name='Y')
X.mybracket(Y).display()

yields

[Y,X] = (-x^2*y + 1) e_x + (-x^3 + 2*x*y^2 - 2*x*y) e_y

and

X.mybracket(Y) == Y.bracket(X)

gives

True

It suffices to attach the function mybracket that you have defined to the vector field class, which is VectorField. You have to import the latter first:

from sage.manifolds.differentiable.vectorfield import VectorField
VectorField.mybracket = mybracket

So the whole code becomes:

from sage.manifolds.differentiable.vectorfield import VectorField
def mybracket(A, B):
     return B.bracket(A)
VectorField.mybracket = mybracket

Then

M.<x,y> = EuclideanSpace()
X = M.vector_field(0,1-y*x^2, name='X')
Y = M.vector_field(-(y-1),x, name='Y')
X.mybracket(Y).display()

yields

[Y,X] = (-x^2*y + 1) e_x + (-x^3 + 2*x*y^2 - 2*x*y) e_y

and

X.mybracket(Y) == Y.bracket(X)
-X.bracket(Y)

gives

True