homework and exercises - Ricci tensor for a 3-sphere without Math packets

Let's have the metric for a 3-sphere:

$$dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right).$$ I tried to calculate Riemann or Ricci tensor's components, but I got problems with it.

In the beginning, I got an expressions for Christoffel's symbols:

$$\Gamma^{i}_{ii} = \frac{1}{2}g^{ii}\partial_{i}g_{ii} = 0,$$

$$\Gamma^{i}_{ji} = \frac{1}{2}g^{ii}\partial_{j}g_{ii},$$

$$\Gamma^{k}_{ll} = -\frac{1}{2}g^{kk}\partial_{k}g_{ll},$$

$$\Gamma^{k}_{lj} = \Gamma^{k}_{lk}\delta^{k}_{j} + \Gamma^{k}_{jk}\delta^{k}_{l} + \Gamma^{k}_{jj}\delta^{j}_{l} = 0.$$

The Ricci curvature must be

$$R_{lj}=\frac{2}{R^{2}}g_{lj}.$$ But when I use definition of Ricci tensor,

$$R_{lj}^{(3)} = \partial_{k}\Gamma^{k}_{lj} - \partial_{l}\Gamma^{\lambda}_{j \lambda} + \Gamma^{k}_{j l}\Gamma^{\sigma}_{k \sigma} - \Gamma^{k }_{l \sigma}\Gamma^{\sigma}_{jk},$$ I can't associate an expression (if I didn't make the mistakes)

$$R_{lj}^{(3)} = \partial_{j}\Gamma^{j}_{lj} + \partial_{l}\Gamma^{l}_{jl} + \partial_{k}\Gamma^{k}_{ll}\delta^{l}_{j} - \partial_{l}\Gamma^{k}_{jk} - \Gamma^{k}_{jk}\Gamma^{j}_{lj} + \Gamma^{k}_{lk}\Gamma^{l}_{jl} + \Gamma^{\sigma}_{k \sigma}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{k}_{jk}\Gamma^{k}_{lk} - \Gamma^{l}_{jl}\Gamma^{j}_{lj} - \Gamma^{l}_{kl}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{j}_{ll}\Gamma^{l}_{jj} - \Gamma^{k}_{ll}\Gamma^{l}_{kl} =$$

$$= \partial_{j}\Gamma^{j}_{lj} + \partial_{l}\Gamma^{l}_{jl} + \partial_{k}\Gamma^{k}_{ll}\delta^{l}_{j} - \partial_{l}\Gamma^{k}_{jk} - \Gamma^{k}_{jk}\Gamma^{j}_{lj} + \Gamma^{k}_{lk}\Gamma^{l}_{jl} + \Gamma^{\sigma}_{k \sigma}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{k}_{jk}\Gamma^{k}_{lk} - \Gamma^{l}_{jl}\Gamma^{j}_{lj} - 2\Gamma^{l}_{kl}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{j}_{ll}\Gamma^{l}_{jj},$$ where there is a summation only on $k, \sigma$, with an expression for the metric tensor.

Maybe, there are some hints, which can help?