Paul Klee never visited the United States, but he played an important role in American art in the 1930s and ’40s. Exhibitions and reproductions of his work, and later translations of his writings, gave American artists permission to do things that weren’t in the shadow of Picasso or Matisse. He seemed a genial figure, and his work was, too, and American artists admired his particular mix of spontaneity and discipline. Klee, it seemed, offered a wealth of visual inspiration, and a way forward, without demanding that anyone wear the straitjacket of a particular style.

“Ten Americans: After Paul Klee” at the Phillips Collection examines the U.S. legacy of the Swiss-born German artist through the work of painters including Jackson Pollock, Robert Motherwell, Kenneth Noland, Gene Davis and Norman Lewis. In some cases, the influence is strikingly obvious, even to the point of outright theft. In other cases, it is more the spirit of Klee than the particulars that has been transmitted. But most of the work on view takes up a small set of visual and creative ideas that were central to Klee’s art: his interest in archaic or “primitive” visual archetypes, the power of the unconscious and his enigmatic, hieroglyphic sensibility.

“Klee was a genius,” the critic Clement Greenberg wrote, before adding, “but he was not a big genius, remarkable as he was, and his influence has been viable precisely because it could not occupy for its exclusive use all of the new territory it opened up.” Greenberg is in bad odor these days for his overbearing influence and dogmatism. But this sums up the show nicely. Klee is a powerful presence, but he leaves room for others, and while it’s easy to admire and even love Klee’s work, it always seems a bit small, not just because he tended to work on a small scale, but because most his paintings are tidy vignettes, putting forth an idea with clarity and charm and occasionally just a hint of something darker. The exhibition curators even quote the museum’s founder, Duncan Phillips, who collected more than a dozen works by Klee, affirming part of this judgment: “Klee builds himself a little house of art in a realm somewhere between childhood’s innocence and everyman’s prospect of infinity.”

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The exhibition includes more than 60 works, many of them clustered in the late 1940s, when abstract expressionism was taking form. With that historical moment lurking in the background, it’s hard to avoid the sense that much of what is going on here is a kind of curtain raiser to that bigger event. That’s unfair, of course, to the work and the artists, who were making these works not as precursors to anything else, but because it was meaningful to them at the time. Mark Tobey, for example, is pushing his images to the edge of the canvas, which is covered in sprays of white tracery. But he is still deploying figures of a sort, discrete things arising from a calligraphic urgency, and generally contained or structured by some overlay. Calligraphic forms by Bradley Walker Tomlin made in the late 1940s float in sumptuously colored grounds, but they aren’t floating freely; there is a sense of frozen motion, a snapshot of an idea, which distinguishes this work from later forms of abstract expressionism.

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Other painters pick up other currents, the biomorphic forms that inspired Joan Miro, cave-painting figures, nervous grids and webbing, and the thick, elastic lines that take on serpentine form, somehow both symbol and picture of the things they seem to represent. Klee was fond of the idea that painting and language were much the same thing, forms of communication that had their origins in basic drawing. But signs are, of course, generally arbitrary, bearing no fixed or obvious relation to the things they represent. And that may be the ultimate weakness not just in Klee’s art but in the art he seemed to liberate in his American admirers: Art based on the invention of symbols can cloy, just like nonsense poetry, as it seems to promise but ultimately denies the possibility of a literal reading.

The next move, to jettison the symbols and figures and invented hieroglyphs along with the old idea that art was meant to be legible, was just around the corner, and already present in works by Pollock. For many viewers, the next stage of that liberation was also much more satisfying.

Ten Americans: After Paul Klee is on view through May 6 at the Phillips Collection. For information, visit phillipscollection.org.