A quaternionic construction of $E_7$

We give an explicit construction of the simply-connected compact real form of the Lie group of type $E_7$, as a group of $28\times 28$ matrices over quaternions, acting on a $28$-dimensional left quaternion vector space. This leads to a description of the simply-connected split real form, acting on a $56$-dimensional real vector space, and thence to the finite quasi-simple groups of type $E_7$. The sign problems usually associated with constructing exceptional Lie groups are almost entirely absent from this approach.