= Real special linear group of degree two
{title2=$SL_2(\mathbb R)$}
= SL2R
{c}
{synonym}
This <Lie group> consists of real two-by-two matrices of determinant one. Its <Lie algebra> is the space of traceless real two-by-two matrices. Its <Exponential map of a Lie group> is not onto: by the <Cayley-Hamilton theorem>, $A^2=-\det(A)I$ for traceless $A$, and its exponential has trace either $2\cosh s$ or $2\cos s$, hence never less than $-2$. The determinant-one matrix $\operatorname{diag}(-2,-1/2)$ is therefore outside its exponential image.
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