I'm studying General Relativity, and I'm currently facing the questions of the metric tensor, the $\Gamma$ connection, and the fact that $\Gamma$ is not a tensor. I'm also reading about the fact that the derivative of a tensor is, in general, not a tensor and in addition to that I'm also reading about the covariant derivative, and here is the question: we have defined the connection in terms of the metric tensor and its derivatives:
$$\Gamma^{\lambda}_{\mu\nu} = \frac{1}{2}\ g^{\lambda\alpha}\left(\frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} + \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}\right)$$
and it's all ok. The covariant derivative has been defined too, and now the professor "proved" that since the normal derivative of a tensor is not a tensor, the covariant derivative will. As a test, he, again, "proved" to run the covariant derivative on the metric tensor, getting:
$$\mathsf{D}_{\lambda} g_{\mu\nu} = \frac{\partial g_{\mu\nu}}{\partial x^{\lambda}} - \Gamma^{\rho}_{\mu\lambda}g_{\rho\nu} - \Gamma^{\rho}_{\nu\lambda}g_{\rho\mu}$$
Ok again, until here. Then "after a little algebra" we find that:
$$\mathsf{D}_{\lambda} g_{\mu\nu} = 0$$
But the "little algebra" is the missing part. Now I would like to prove it, but I'm stuck on the calculation and I'm going to tell you how and why.
First of all, let me tell you that I started with substituting the $\Gamma$, getting the long expression:
\begin{equation*} \begin{split} \mathsf{D}_{\lambda} g_{\mu\nu} & = \frac{\partial g_{\mu\nu}}{\partial x^{\lambda}} - \bigg\{\frac{1}{2}\ g^{\rho k}\left(\frac{\partial g_{\mu k}}{\partial x^{\lambda}} + \frac{\partial g_{\lambda k}}{\partial x^{\mu}} - \frac{\partial g_{\mu\lambda}}{\partial x^{k}}\right)\bigg\}g_{\rho\nu} \\\\ & - \bigg\{\frac{1}{2}\ g^{\rho \alpha}\left(\frac{\partial g_{\nu \alpha}}{\partial x^{\lambda}} + \frac{\partial g_{\lambda \alpha}}{\partial x^{\nu}} - \frac{\partial g_{\lambda\nu}}{\partial x^{\alpha}}\right)\bigg\}g_{\rho\nu} \end{split} \end{equation*} \ \ Then I continued to develop the calculation but here is the problem : whilst developing the calculation, I met lots of terms of this form (for example the very fist one):
$$\frac{1}{2} g^{\rho k}\frac{\partial g_{\mu k}}{\partial x^{\lambda}}\ g_{\rho\nu}$$
There are other five terms like this, with different indexes of course, but I don't know how to proceed. Tensor calculus when it comes to manipulation of index is somehow "strange". I never know what to do because nobody taught us anything about if not the basics. And when the professor says "after little algebra" I would like to see that little algebra at least once in a lifetime, right?
So, any hint / explanation / something for this? My problem is how to treat objects like that. There are too many indexes, and I need to really understand how to proceed and work. Thank you for the time!