How to calculate some Ricci tensor derived quantities.

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I am unsure of the correct way to calculate some Ricci tensor derived quantities, I have tried many different ways but can't get the correct value for the two independent Bach tensor parts U and V (W is zero in four dimensions).

The quantities I am unsure of are labeled "Unsure" in the code below.

var('tpar','lpar')

tpar = 4
lpar = 4

Parallelism().set('tensor',tpar)
Parallelism().set('linbox',lpar)
show(Parallelism())

var('a')

%display latex

viewer3D = 'threejs'  # must be 'threejs', 'jmol', 'tachyon' or None (default)

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')

MCT = M.open_subset('MCT')  # cartesian
CT.<t,x,y,z> =  MCT.chart(r't:(-oo,+oo) x:(-oo,+oo) y:(-oo,+oo) z:(-oo,+oo)')

MBL = M.open_subset('MBL')  # Boyer-Lindquist
BL.<t,r,th,phi>   = MBL.chart(r't:(-oo,+oo) r:(0,+oo) th:\theta:(0,+pi):periodic phi:\phi:(0,2*pi):periodic) ') 
M.declare_union(MBL,MCT)

g = MBL.lorentzian_metric('g')
var('r', 'th')
g00 = -e^(2*r)
g11 = e^(-2*r)
g22 = e^((th+r))
g33 = 0
g03 = 1
g[0, 0] = g00
g[1, 1] = g11
g[2, 2] = g22
g[3, 3] = g33
g[0, 3] = 1/2*g03

Nabla = g.connection()

# http://liu.diva-portal.org/smash/get/diva2:244008/FULLTEXT01.pdf equations 4.52 4.53

v1 = -Nabla(Nabla(g.ricci_scalar()))                           # Unsure
v2 = +(g*(Nabla(Nabla(g.ricci_scalar())))).up(g, 3)['^a_bca']  # Unsure
v3 = -g.ricci_scalar()*g.ricci()                                 
v4 = +1/4*(g*g.ricci_scalar()*g.ricci_scalar())  

V = v1 + v2 + v3 + v4
V[:]

u1 = +Nabla(Nabla(g.ricci())).up(g, 3)['^a_bca']                # Unsure
u2 = -1/2*v2
u3 = +2*((g.weyl().down(g))*(g.ricci().up(g)))['^ad_abcd']
u4 = +2*(g.ricci()*(g.ricci())).up(g, 3)['^a_abc']    
u5 = -1/2*(g*g.ricci()*(g.ricci().up(g)))['^ad_bcad']
u6 = -1/3*(g.ricci_scalar()*g.ricci())
u7 = +1/12*(g*g.ricci_scalar()*g.ricci_scalar())

U = u1 + u2 + u3 + u4 + u5 + u6 + u7

show(' U/2: ', 1/2*U[:], ' V/6: ', 1/6*V[:])

B = (1/2*U)['_bc']+(1/6*V)['_bc']
show('B: ', B[:])
Bach = Nabla(g.cotton()).up(g,3)['^u_aub'] + ((g.schouten().up(g))*(g.weyl().down(g)))['^uv_aubv']
show('Bach: ', Bach[:])

The correct Bach tensor (I have altered the cotton tensor 'convention') is:

[    1163/32*e^(6*r)                   0                   0       25/64*e^(4*r)]
[                  0       -5/32*e^(2*r)                   0                   0]
[                  0                   0 -45/32*e^(5*r + th)                   0]
[      25/64*e^(4*r)                   0                   0                   0]