$$\large R_{ij} = R^{\ell}_{i\ell j} = g^{\ell m}R_{i\ell jm} = g^{\ell m} R_{\ell imj} = \frac{\partial\Gamma^{\ell}_{ij}}{\partial x^{\ell}} - \frac{\partial \Gamma^{\ell}_{i\ell}}{\partial x^j} + \Gamma^{\ell}_{ij}\Gamma^m_{\ell m} - \Gamma^m_{i\ell}\Gamma^{\ell}_{jm} $$
I'm fairly good with tensors but I don't understand the above equation. In the equation for the Ricci curvature in terms of the Christoffel symbols, there are two terms that appear to be "products" of two Christoffel symbols (please correct if wrong) and I want to know what is stopping the terms from cancelling each other out. I mean, the last two terms on the extreme right hand side of the equation, there are those two terms that look like they're products of Christoffel symbols.
After a contraction (I think) the indices $i$ and $j$ are left out. My question is, can those last two terms be cancelled out based on some symmetry or ant-symmetry condition (even though Christoffel symbols are not tensors)?
The part of the expression you are referring to is $\Gamma^{\ell}_{ij}\Gamma^m_{\ell m} - \Gamma^m_{i\ell}\Gamma^{\ell}_{jm}$.
Suppose now that the dimension is two so that each index takes the value $1$ or $2$. Fixing $i$ and $j$, we have
\begin{align*} \Gamma^{\ell}_{ij}\Gamma^m_{\ell m} - \Gamma^m_{i\ell}\Gamma^{\ell}_{jm} &= \sum_{\ell=1}^2\sum_{m=1}^2\Gamma^{\ell}_{ij}\Gamma^m_{\ell m} - \Gamma^m_{i\ell}\Gamma^{\ell}_{jm}\\ &= \sum_{\ell=1}^2\Gamma^{\ell}_{ij}\Gamma^1_{\ell 1} - \Gamma^1_{i\ell}\Gamma^{\ell}_{j1} + \Gamma^{\ell}_{ij}\Gamma^2_{\ell 2} - \Gamma^2_{i\ell}\Gamma^{\ell}_{j2}\\ &= \Gamma^1_{ij}\Gamma^1_{11} - \Gamma^1_{i1}\Gamma^1_{j1} + \Gamma^1_{ij}\Gamma^2_{12} - \Gamma^2_{i1}\Gamma^1_{j2} + \Gamma^2_{ij}\Gamma^1_{21} - \Gamma^1_{i2}\Gamma^2_{j1} + \Gamma^2_{ij}\Gamma^2_{22} - \Gamma^2_{i2}\Gamma^2_{j2}. \end{align*}
For $i = 2, j = 2$ the expression becomes
\begin{align*} &\ \Gamma^1_{22}\Gamma^1_{11} - \Gamma^1_{21}\Gamma^1_{21} + \Gamma^1_{22}\Gamma^2_{12} - \Gamma^2_{21}\Gamma^1_{22} + \Gamma^2_{22}\Gamma^1_{21} - \Gamma^1_{22}\Gamma^2_{21} + \Gamma^2_{22}\Gamma^2_{22} - \Gamma^2_{22}\Gamma^2_{22}\\ =&\ \Gamma^1_{11}\Gamma^1_{22} - \Gamma^1_{12}\Gamma^1_{12} + \Gamma^1_{22}\Gamma^2_{12} - \Gamma^1_{22}\Gamma^2_{12} + \Gamma^1_{12}\Gamma^2_{22} - \Gamma^1_{22}\Gamma^2_{12} + \Gamma^2_{22}\Gamma^2_{22} - \Gamma^2_{22}\Gamma^2_{22}\\ =&\ \Gamma^1_{11}\Gamma^1_{22} - \Gamma^1_{12}\Gamma^1_{12} + \Gamma^1_{12}\Gamma^2_{22} - \Gamma^1_{22}\Gamma^2_{12}. \end{align*}
In general, this combination need not be zero. For example, if we take the unit sphere $S^2$ with the metric $g = d\theta\otimes d\theta + \sin^2\theta\, d\phi\otimes d\phi$, the only non-zero Christoffel symbols are
$$\Gamma^1_{22} = -\sin\theta\cos\theta\qquad \text{and}\qquad \Gamma^2_{12} = \Gamma^2_{21} = \frac{\cos\theta}{\sin\theta}.$$
So in this case, the expression becomes
$$\Gamma^1_{11}\Gamma^1_{22} - \Gamma^1_{12}\Gamma^1_{12} + \Gamma^1_{12}\Gamma^2_{22} - \Gamma^1_{22}\Gamma^2_{12} = 0 - 0 + 0 - (-\sin\theta\cos\theta)\left(\frac{\cos\theta}{\sin\theta}\right) = \cos^2\theta \neq 0.$$