= Compact real form of a complex semisimple Lie algebra
{title2=$\mathfrak u\otimes_{\mathbb R}\mathbb C\cong\mathfrak g$}
= Compact real form
{synonym}
A compact real form of a complex <semisimple Lie algebra> $\mathfrak g$ is a real <Lie subalgebra> $\mathfrak u$ whose <complexification of a Lie algebra> is $\mathfrak g$ and whose <Killing form> is negative definite. Its simply connected <Lie group> is compact. The positive internal <inner product> used in unitary gauge theory is therefore $-\kappa$. For $\mathfrak{sl}_2(\mathbb C)$ the <SU(2) Lie algebra> is a compact real form, whereas $\mathfrak{sl}_2(\mathbb R)$ is a different real form with indefinite Killing form.
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