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  "body": "\n## Summary\n\nThe Whitehead product is an operation $[\\alpha, \\beta] \\in \\pi_{p+q-1}(X)$ on [[homotopy groups|P976WYEFMB]], defined for $\\alpha \\in \\pi_p(X)$ and $\\beta \\in \\pi_q(X)$.\nIt comes from the attaching map of the top cell of the product $S^p \\times S^q$.\nFor $p = q = 1$ it is the commutator in the fundamental group.\nWith it, the homotopy groups of a space form a graded Lie algebra, up to a shift of degree and signs.\n\n## Explanations\n\n### L1 human\n\nThe product of two spheres $S^p \\times S^q$ is built from the wedge $S^p \\vee S^q$ by gluing in one more cell.\nThe gluing map is a sphere of dimension $p + q - 1$ sitting in the wedge.\nGiven two spheres in a space $X$, sending that gluing map into $X$ gives a new element of a higher homotopy group: their Whitehead product.\nFor two loops on a torus the product is trivial, which reflects that the two loops commute there.\n",
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