When Martha Argerich last appeared at the Proms with Daniel Barenboim and the West-Eastern Divan Orchestra three years ago, she played Liszt’s First Piano Concerto, a work that had not been part of her regular repertory for many years. For her return visit with her old friend and his orchestra, Argerich opted for another work that has not figured in her London appearances in a long time, Tchaikovsky’s First Piano Concerto.

As always with Argerich, it was a mesmerising performance. Not a perfect one by any means – some of her characteristic changes of gear, especially in the finale, left Barenboim and the orchestra scrambling to keep up. Nor was it a particularly profound or searching one, but it bristled with intensity and intent from the first bar, so often propelled by the crystal-clear precision that still comes so naturally to her, and the lines of the slow movement floated with weightless ease.

There was no encore this time from Argerich, but at the end of the concert Barenboim and the orchestra did follow Witold Lutosławski’s Concerto for Orchestra with Beethoven’s Egmont Overture. By then, though, there had already been plenty of proof that WEDO, celebrating its 20th birthday this year, is now a very fine band. The woodwind and brass are particularly characterful, and Barenboim used them to bring out the dark, threatening undertow to Lutosławski’s early showpiece, as well as its elements of unbuttoned virtuosity. Schubert’s Unfinished Symphony had been immaculate, too, viewed very much through the prism of late-19th-century romanticism, with long, Bruckner-like tendrils of melody. Not to all tastes, but effective in this context, at least.

There are pianists, and then there’s Martha Argerich. Teaming up once again with Daniel Barenboim, who seems to come alive when he’s conducting the West-Eastern Divan Orchestra, this pianistic force of nature brought to the eager Proms audience a piece that was once purportedly declared bad and vulgar by pianist Nikolai Rubinstein, but has become one of the most popular concertos of all time and a mainstay of Argerich’s repertoire.

Tchaikovsky’s Piano Concerto no. 1 in B flat minor is a conundrum. Some say the first movement is a hotchpotch; others say it is innovative. Either way, in Argerich’s hands it is a lyrical piece, full of extrovert virtuosity and with melodies aplenty but also subtle and intimate. In this performance, Argerich showed granite determination and magical delicacy, and while the forceful first movement was more deliberate than reckless, this did reveal some fine detailing, particularly in the seamless interplay between soloist and orchestra. Vivacious and precise, Argerich had the heart of a lion in the third movement but, despite the bluster and technical wizardry of the outer movements, if you really want to hear how this concerto should be played, the second movement was the one for me. Argerich’s mercurial shimmerings and the gentle ebbing of the orchestra revealed subtleties and voices that are sometimes lost, and Argerich defied logic with the apparent simplicity of her playing. Yes, there were one or two scuff marks, but this was a very polished performance indeed.

In the orchestra's 20th anniversary year, the choice of concert opener was a piece that started their Proms journey back in 2003. Schubert’s Symphony no. 8 (the “Unfinished”) is a masterpiece of tension and release, and Barenboim’s brooding first movement allowed him to build up a fine sense of drama, although the rather slow pace resulted in a lack of bite and some over-smoothing, before caressing the Royal Albert Hall with a calm air of serenity in the Andante con moto, revealing wonderfully cultured horns and a clean purity of tone in the woodwinds. The string playing was lustrous and fulsome throughout, with Barenboim’s keen feel for the shape of the piece and his carefully placed changes in dynamics winning the day.

Witold Lutosławski’s celebrated Concerto for Orchestra was the piece that put him firmly on the map. Written at the end of his folk-inspired period before his migration towards modernism, it shows through its dynamism and depth not only the composer’s great sense of structure and invention but also how gifted he was as an orchestrator. Barenboim clearly has an affinity with this piece, and it showed in his determination to draw out both precision and cohesion. The Intrada was sharp and gutsy, the orchestra producing a big, bold sound and plenty of rhythmic drive (shades of Bartók), and the muted chatterings of the nocturnal Capriccio had sprinkles of sparkle, with the sinister procession of the Passacaglia in the third movement leading to the gorgeous harmonies of the Corale section and the frantic climax being both terrifying and ecstatic. Lutosławski described his music as “fishing for souls”, looking for listeners who felt the same way as he did. With a performance of this versatility and conviction, I wouldn’t be surprised if this gets you hooked.

Barenboim rarely gets away without an encore, and the drama and drive of Beethoven’s Egmont Overture gave this Proms audience exactly what they wanted.