Throughout this review, \(F\) is a field of characteristic different from \(2\), \(A\) is a simple \(F\)-algebra (though the authors sometimes work more generally), and \(\sigma\) is an involution on \(A\).
Roughly following Baer, an ordering of \(F\) is a maximal subset \(P \subseteq F\) of \(\sigma\)-symmetric elements (i.e., \(\sigma(x) = x\) for all \(x\in P\)) satisfying the following properties:

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\(0\in P\),

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\(P + P \subseteq P\),

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\(\sigma(a)Pa \subseteq P\) for all \(a\in F\),

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\(P\cap -P = \{0\}\).

In the case when \(\sigma\) is \(F\)-linear, the authors define a positive cone over \(P\) on \(A\) to be a maximal subset \(\mathcal{P}\subseteq A\) satisfying all the same properties as above, and additionally such that \(\{u\in F : u\mathcal{P} \subseteq \mathcal{P}\} = P\). These behave similarly to orderings in many ways: in particular, a positive cone \(\mathcal{P}\) induces a partial ordering \(\leq_{\mathcal{P}}\) on \(A\) which respects addition.
A valuation on \(F\) is a function \(v: F\to \Gamma_v\cup\{\infty\}\) satisfying the following for all \(x,y\in F\):

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\(v(x) = \infty \Leftrightarrow x = 0\),

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\(v(x+y) \geq \min\{v(x), v(y)\}\),

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\(v(xy) \geq v(x) + v(y)\).

A \(v\)-gauge on \(A\), in the sense of Tignol and Wadsworth, is a map \(w: A\to \Gamma\cup\{\infty\}\) (where \(\Gamma_v\subseteq \Gamma\)) satisfying the same properties as above, and additionally such that \(w(\lambda x) = v(\lambda) + w(x)\) for all \(\lambda\in F, x\in A\), that \(A\) has an \(F\)-basis \(\{e_1, \dots, e_m\}\) such that \(w(\lambda_1 e_1 + \dots + \lambda_m e_m) = \min\{w(\lambda_i e_i)\}\) for all \(\lambda_i\in F\), and that \(\mathrm{gr}(A)\) is a semisimple \(\mathrm{gr}(F)\)-algebra.
It is already known that, given an ordering \(P\) on \(F\) and a subfield \(k\) of \(F\), it is possible to define a valuation \(v_{k,P}\) on \(F\): indeed, \(R_{k,P} := \{x\in F : \sigma(x)x \leq m \text{ for some } m\in k\}\) is a valuation ring of \(F\). Again reverting to the case when \(\sigma\) is \(F\)-linear, the authors show that, given a positive cone \(\mathcal{P}\) over \(P\) on \(A\), there is a unique \(\sigma\)-invariant \(v_{k,P}\)-gauge \(w = w_{k,\mathcal{P}}\) on \(A\), and it satisfies \(w(\sigma(x)x) = 2w(x)\) for all \(x\in A\) (Theorem 5.16).
The valuation \(v\) and the ordering \(P\) are said to be compatible if, for all \(x,y\in F\), \(0\leq_P x\leq_P y\) implies \(v(x) \geq v(y)\). The authors extend this and many other equivalent compatibility conditions to compatibility conditions between gauges and positive cones (§6), and show that they are all equivalent (Proposition 6.7). In the remainder of the paper, they prove lifting results for these simple rings \(A\) related to the classical Baer-Krull theorem for \(F\) (Theorem 8.9).