Brahms wrote only three cello-based compositions in his lifetime - two solo sonatas, and finally a double concerto for violin and cello in A minor, written in 1887.
Brahms wrote four concertos during his lifetime, and this Double Concerto in A Minor is the last of them; It was also his last published orchestral work. It was originally commissioned by cellist Hausmann for cello concerto, when Brahms was on holiday in Switzerland and had intended to write his Symphony No. 5. After Hausmann's invitation, he started thinking of composing a double concerto for violin and cello.
In general, concertos are for a solo instrument plays with an orchestra. In contrast, double concerto is, as the name suggests, a concerto in which two instruments play together with an orchestra. This kind of work was rarely composed in European music circles at the time. On the other hand, Brahms incorporated some of the concepts of the never-seen Symphony No. 5 into this concerto. Therefore, the work still sounds like a symphony at first glance, with its complex arrangements and majestic atmosphere.
The Double Concerto in A minor has a typical classical structure. Tth three movements move from Allegro to Allegro, ending with Vivace non troppoe. Most of the Brahms lovers would have heard of this piece and the behind story that he was using this work to seek for a re-union with the violinist Joachim. He enclosed the composition in the letter to the violinist who later accepted his apology and became the violin soloist at the premiere of the work.
Indeed, the arrangement of the work contains elements that Brahms added specifically for Joachim. The melodies of both the cello and violin are in rondo form which is common in Hungarian dance music, given that Joachim was a Hungarian and familiar with such genre. Besides, he made use of the musical motif A–E–F, a permutation of F-A-E for the whole piece. This is a tribute to Joachim's motto, Frei aber einsam ("free but lonely"). Brahms was involved in another tribute work to Joachim when he was much younger, and this concerto again showed his appreciation of the violin's motif.
To many critics, it was a bold choice to match violin and cello, since each of them has very distinctive tone color and style. Indeed, this work was not well received after its premiere. He was criticised by his long term companion Clara Schumann for "not being brilliant for the instruments", and by the musicologist and biographer Richard Specht for "being one of Brahms' most inapproachable and joyless compositions. Yet it was commenting positively afterwards and now it is one of the masterpieces of the Double Concerto in classical music today!
Music reference: https://www.youtube.com/watch?v=5OjLKhmzQTA&t=1780s