  The Euler class characterizes the topology of two real bands isolated from
other bands in two-dimensions. Despite various intriguing topological
properties predicted up to now, the candidate real materials hosting electronic
Euler bands are extremely rare. Here, we show that in a quantum spin Hall
insulator with two-fold rotation $C_{2z}$ about the $z$-axis, a pair of bands
with nontrivial $\mathbb{Z}_2$ invariant turn into magnetic Euler bands under
in-plane Zeeman field or in-plane ferromagnetic ordering. The resulting
magnetic insulator generally carries a nontrivial second Stiefel-Whitney
invariant. In particular, when the topmost pair of occupied bands carry a
nonzero Euler number, the corresponding magnetic insulator can be called a
magnetic Euler insulator. Moreover, the topological phase transition between a
trivial magnetic insulator and a magnetic Stiefel-Whitney or Euler insulator is
mediated by a stable topological semimetal phase in which Dirac nodes carrying
non-Abelian topological charges exhibit braiding processes across the
transition. Using the first-principles calculations, we propose various
candidate materials hosting magnetic Euler bands. We especially show that
ZrTe$_5$ bilayers under in-plane ferromagnetism are a candidate system for
magnetic Stiefel-Whitney insulators in which the non-Abelian braiding-induced
topological phase transitions can occur under pressure.
