Consider the local rings

$$R = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw\rangle$$

and

$$S = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw+xyzw\rangle.$$

Is $R$ isomorphic to $S$?

**Some context**:  I am trying to understand formal neighborhoods of points on certain varieties.  I expect one answer, and I'm getting a different answer.  This is the first nontrivial case where the answer that I get does not obviously agree with the answer that I expect.

**Some history**:  In a previous post (<https://mathoverflow.net/questions/186886/two-rings-are-they-isomorphic>), I asked a version of this question with one fewer variable, and Bjorn Poonen pointed out that the rings are isomorphic because there is only one kind of rational double point.  I think this was essentially an accident, hence the new post.