How to handle convos about recently deceased people in Japanese?

Hello guys,

So as stated in the Title: how does one handle a conversation in Japanese in which the topic shifts to one of my family members how has died recently?

As ask this Question because I've heard and observed from my own experiences that many Japanese people can be very indirect or even avoiding with these kinds of topics.

For example: I've heard from a Japanese man (who's cat has died recently) that , and I quote, "maybe my cat isn't too well, it may be dead even"

From my personal experience, most Japanese people don't talk about topics such as these unless their genuinely close to you on a personal level and way past that "Smalltalk level".

Of course, mentioning that a family member of mine has recently deceased and how painful it has been for me isn't something I would mention to begin with, but if one would partake in a deeper conversation and wanted to mention this, how would/should they do it?

In a way that would be culturally acceptable and show respect to all parties included…