Hi everyone!

I have a problem at hand that needs to calculate a series of covariant derivatives like

```
\nabla_{\alpha} \nabla_{\beta} \nabla_{\gamma} \nabla_{\lambda}\partial_{\nu}\phi
```

Where \nabla is the usual covariant derivative in GR and \partial is the usual partial derivative.

In fact, this expression is a term of an infinite series, thematically

```
g^{\mu \nu} \partial_{\mu}\phi (\sum\limits_n \nabla^{2n}) \partial_{\nu}\phi
```

where I am only interested in the some first terms of the series as follow

```
(\nabla)^2 = g^{\alpha \beta} \nabla_\alpha \nabla_\beta
(\nabla)^4 = g^{\alpha \beta} g^{\mu \nu} \nabla_\alpha \nabla_\beta \nabla_\mu \nabla_\nu
(\nabla)^6= ...
```

Here \nabla is the usual covariant derivative for a covector.

```
\nabla_\alpha \partial_{\beta}\phi = \partial_{\alpha \beta}\phi - \Gamma^\lambda_{\alpha \beta} \partial_{\lambda}\phi
```

where \Gamma represents (symmetric) Christoffel symbols.

I have read the the excellent paper written by Leo Brewin but the way that he implements the covariant derivative is not clear for me and I can't generalize it to my case.

Can anyone, please, give me a hint how to implant this expression in Cadabra?