When living in the US, I noticed that there were no differences between cultures when it comes to the loss of someone: people are affected by the loss, and it takes time to heal. During the mourning process, people can be sad, tired, depressed and so on... Expressing can be very different though. Many european countries I know have the same "etiquette", and it seemed to me that they are very similar to the ones in the US.
In the case of death, when consolation is not possible, just to let the bereaved know that you care is doing your best. Offer your own words in the most considerate, genuine, kind, and courteous manner.1
Also said, keep in mind the "3 C rule" : Compassion, Courtesy, Common sense. Show that you care with an expression of sorrow, and keep it as simple as possible. Beware of:
- unsolicited advice.
- words meant to “cheer up” or encourage.
- words of clichéd wisdom or of your own experience.2
I've been taught that you should say or write very very little, and be very careful not to do too much. The worst I've ever heard was : "I know how you feel...". No, you don't! No one knows, we're all different. We have suffered the same thing, it doesn't mean we feel or react the same way.
In your case, nothing personal, keep it "professional", so very very simple, on a card or note:
I'm so sorry to learn about [family member]. If there's anything I can do, please let me know.
I'm not writing emails because I think there's nothing appropriate you can write in the "object" case. I send a simple card. You can send this to his office.
Even if this is affecting your work, and you struggle, look for alternatives measures, channels or strategies, avoid bothering the person who just lost someone (or is dealing with serious illness).
Even when attending a funeral, I would not go beyond this. It can be said or written, I've always done that for the ones that I knew (or not), family, coworkers, relatives or acquaintances.
Always keep it short and simple, don't add anything to their pain.
1 2 Candace Smith Etiquette -- Emily Post Etiquette -- Indeed / message of condolence (because it's almost a professional background in your case, so examples can be useful)