The attack prompted the French government to up its terror alert to "attack emergency", the highest of its three warning levels, the embassy said.
The suspected gunman who attacked a market in the French city of Strasbourg is dead, police sources have told Reuters.
He was said to have entered central Strasbourg via the Corbeau bridge to the south of the city centre, before heading to Rue des Orfevres, a popular shopping street close to the cathedral, where he opened fire at around 8pm local time (7pm GMT).
Shortly after the shooting, lines of police vehicles and ambulances streamed into the market area, under festive lights declaring the city the "capital of Christmas".
He was not at home, but police found a grenade, four knives and a pistol.
French investigators call them the "gangster-jihadists" - young men, often from poor immigrant backgrounds, who start with petty crime, drug dealing and robbery and graduate to terrorism...
While authorities urged people in the area to stay inside after Tuesday's attack, Strasbourg Mayor Roland Ries told BFM television Wednesday that "life must go on" so that the city does not cede to a "terrorist who is trying to disrupt our way of life".
He was placed in custody on Thursday morning; the four others detained were Chekatt's parents and two of his brothers.
The 29-year-old man being sought by police over a mass shooting in Strasbourg lived in a small apartment in a rundown housing bloc and has spent time in prison in France and Germany.
Chekatt escaped from the scene in a taxi, bizarrely bragging to the driver about being injured in a firefight with soldiers after killing 10 people, officials said.
Chekatt was subject of a "fiche S", literally an S card, which is used by police to flag someone who is considered a threat to national security. The latest person to be arrested is a member of his "entourage".
Chekatt fired at least three volleys at the market crowds, then engaged in at least two seperate exchanges of fire with patrolling soldiers-put onto the streets as part of a nationwide security operation introduced after a series of terrorist attacks in Paris in 2015. Three people were killed and several others were injured. "They, too, feared being publicly named because the gunman is still being hunted by police". "I only learned Anuphong was killed by stray bullets", his cousin Thiraphon Chatchayaworachat said on Wednesday. "I heard gunshots outside".
The European Parliament president, however, says that legislative sessions will continue for the rest of the week despite the attack and a lockdown of the building.
AP journalists and a French police official say a police operation is going on in the Strasbourg neighborhood where a suspected Christmas market gunman was last seen.