Brown served in an at-large Caddo Parish delegation during his entire House career. By the time that he declined to seek a seventh term in 1972, single-member districts were instituted in Louisiana legislative races. In the March 3, 1964, general election, Brown ran third for the five available seats but was outdistanced by two Republican (GOP) candidates, Morley A. Hudson and Taylor W. O'Hearn. Joining Brown in the delegation were fourth-place finisher Frank Fulco and newcomer J. Bennett Johnston, Jr., later a United States senator, who narrowly grabbed the fifth slot over fellow Democrat Wellborn Jack, another Shreveport attorney and long-term legislative member. Brown said that he was elated to win his fifth term but was not too pleased at finishing behind two Republicans, who had benefited from the coattails of Shreveporter Charlton Lyons, who was waging the first well-organized GOP campaign for governor in modern Louisiana history.
In 1960, Brown co-sponsored the bill which created the former ten-member Joint Legislative Committee on Un-American Activities, which was modeled after the national House Committee on Un-American Activities and similar bodies in other states. The stated purpose of the committee was to investigate "communist and socialist activities" within Louisiana. The bill achieved final approval but only after the Louisiana State Senate amendeded it to require that the committee act through the office of the state attorney general, then Jack P.F. Gremillion, to enforce contempt actions.
Brown was born to John Spence Brown and the former Melody Bryan in a log house on a farm near Waldo in Columbia County near Magnolia in southwestern Arkansas. One of seven children, he outlived his six siblings. The Browns moved to Shreveport in 1924, where Algie graduated in 1928 from C.E. Byrd High School, the first public high school in the city. One of his Byrd classmates was his future state legislative colleague Frank Fulco, who became a leader of the Italian American community in Louisiana. Thereafter, in 1934, Brown received a bachelor of arts degree from the Methodist-affiliated Centenary College.
In 1935, Brown obtained his law degree from Louisiana State University in Baton Rouge. That same year, he established his law practice, which was interrupted after eight years, by World War II. Brown was a United States Navy lieutenant aboard several aircraft carriers in the Pacific Theater. He was a radar control officer aboard the USS Natoma Bay when the escort carrier was struck by a Japanese kamikaze airplane during the Okinawa campaign in June 1945. Brown was discharged from active duty in 1946 and resumed his law practice. In 1947, Brown wed the former Hazel Turner (April 29, 1919 - June 21, 1994). In 1996, he married the former Elise Beaudreaux (December 20, 1923 - October 1, 2003) of Ruston, the seat of Lincoln Parish in north Louisiana. Brown died at his Shreveport home after a lengthy illness. Services were held in the Frost Chapel of the First Baptist Church of Shreveport, of which Brown had been a member for seventy-five years. Brown was buried beside first wife Hazel at Forest Park Cemetery in Shreveport. He was survived by two sons, Curtis Brown of Shreveport and Bryan Brown and wife, Susan Jones Brown, then of Salt Lake City, Utah, two grandsons, and three stepchildren.