The Riemann tensor can be thought of as a map from a triple of vectors X, Y , Z to a fourth vector, written in a coordinate-free way as
$R(X, Y )Z = \nabla_X \nabla_Y Z − \nabla_Y \nabla_X Z − \nabla_{[X,Y ]}Z\tag 1$
where the notation $\nabla X = X^\mu\nabla_{\mu}$ refers to the covariant derivative along the vector X, so that (1) is equivalent to
$R^ρ_{σμν} X^μY^νZ^σ = X^λ∇_λ(Y^η∇_ηZ^ρ) − Y^λ∇_λ(X^η∇_ηZ^ρ) −(X^λ∂_λY^η − Y^λ∂_λX^η)∇_ηZ^ρ \tag 2$
How do I show that this is equivalent to the usual definition of the Riemann tensor in terms of Christoffel symbols?
That is, to:
$R^a_{bcd}X^cY^dZ^d=X^c\partial_c(\Gamma^a_{db})Y^dZ^b-Y^d\partial_d(\Gamma^a_{cb})Y^cZ^b+ \Gamma^a_{cm}\Gamma^m_{db}X^cY^dZ^b-\Gamma^a_{dm}\Gamma^m_{cd}X^cY^dZ^b \tag 3$
I 've trying for a while but I getting lost with indices