Thanks for trying. I had no other Firewire devices connected. Every piece of hardware and software have been upgraded. I combed the net in a search for answers, but could not find any.
I contacted Lacie Support. They gave me some advises which I folllowed with no results, but they had no real solution for me suggesting sending the unit back to them (at my cost). I decided to take my burner plus my G4 to the Westworld Macintosh Tech Support. No results. The burner DID work on other machines, but not on mine. Other Firewire devices were working with my machine, but not my newly bought Lacie DVD burner. They tried a new, just from the box, Lacie burner, but it did not work on my G4 either.
They also contacted Lacie Support. The Lacie Support team has not heard of such problem. Finally, They decided to install the internal LaCie d2 DVD-RW (My G4 had a slot for that) — it worked. They took back the external burner — and that was it for me.
I am not going to find out what was wrong when even 2 teams of tech experts could find out.
Thanks again for your advise.