Let $G$ be the abolute Galois group of $\mathbb Q_p$, let $\delta_1, \delta_2: G\rightarrow L^{\times}$ be continuous characters, where $L$ is a finite extension of $\mathbb Q_p$. Assume that $\delta_1\delta_2^{-1}$ is neither trivial nor the cyclotomic character then $Ext^1_{G}(\delta_2, \delta_1)$ is one dimensional. Hence there exists a unique non-split extension:

$0\rightarrow \delta_1\rightarrow V\rightarrow \delta_2\rightarrow 0$.

When is $V$ de Rham?

I believe that the answer is if and only if both $\delta_1$ and $\delta_2$ are de Rham and the Hodge-Tate weight of $\delta_1\delta_2^{-1}$ is $\ge 1$ (at least if the Hodge-Tate weights of $\delta_1$ and $\delta_2$ are distinct) and I guess I could x it out by using Bloch-Kato's paper in Grothendieck Festschrift, bu the answer must be well known and maybe even written down somewhere.

Ideally, I would like to be able to quote a reference, where this has been worked out.